aub  &!)oms0n's  Sctie0. 

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(I  r  Ii  A  •;  T  I  C  A  L 

'     *. 

\\  ARITHMETIC, 

f-      I 

KriC  ^  •  TKITINO  /iis 

iDUC" 

ALSO,   JLLTISTRATINO    THE     -4^^- 

PRINCIPLES.  VTION. 


FOR  SCHOOLS  £KD  ACADE? 


By  JAMES   r,. 


L,  » JL    AUTHOR  OF  MKVTAl  AKnji.HKTir  ;  t*f< 
:W-1:  nia-aw*  AR1T!IMK*' 


1 
• 


r> 


NSW    YORK: 

MARK  H.  NEWMAN    &    CO.,    199   BROADWAY. 
:  o  i  :;  N  A  T  i :    w  .    n .    MOORE    A    c  o . 
CHICAGO:  CHICKS,  BROSS  <t  CO. 
•  UN:  ivisoN  A  co. 


Kennebec  Historical  Society. 


From  the  Library  of 


Harlow  Spaulding  of  Hallowell, 
and  of  Hammonton,  New  Jersey. 

January,  1901. 


ARITHMETIC, 


-& 


WITH  THE  SYNTHETIC  MODE  OF  INSTRUCTION 


PRINCIPLE'S  OF  CANCELATION. 


BY  JAMES  B.  THOMSON,  A.M., 

UJTHOR  OF  MENTAL  ARITHMETIC  J  EXERCISES  IN  ARITHMETICAL  ANALYflMrf  ^ 
HIGHER  ARITHMETIC  ;  EDITOR  OF  DAY'S  SCHOOL  ALGEBRA; 
LKOENDUB'S  GEOMETRY,  ETC. 

I 

SIXTY  EIGHTH  EDITION,  REVISED  AND  ENLARGED* 


NEW  YORK: 

PUBLISHED  BY  MARK  H.  NEWMAN  &  CO., 
199    BROADWAY. 
1851. 


DAY  AND       OMSON'S  MATHEMATICAL  SERIES 

AND   ACADEMIES 


I.  MENTAL  ARITHMETIC;  or,  First  Lessons  in  Numbers  ;- 
For  Beginners.  This  work  commences  with  the  simplest  combinations 
of  numbers,  and  gradually  advances  to  more  difficult  combinations, 
d^theHninoLof  theiiearner  expands  and  is  prepared  to  comprehend 

thera-  i!  >L  ^  V\  .     i 

II.  FkAOTICAL    ARITHMETIC;—  Unitih-g    the.    Inductive 
with  f.he  Synthetic  mode  of  Instruction  ;  also  illustrating-^he  prin- 
ciples of  CANCELATION.     The  design  of  this  work  is-to  make  the 
pupil  thoroughly  acquainted   with    the  reason  of  every  operation 
which  he  is  required  to  perform.     It  abounds^  in  examples,  and  is 
emi'nently  practical. 

1*m.«E^TJp\  PRACTICAL  'ARITHMETIC  ;—  Containing 
tlie"  answers^wiin  rtumerous  suggestions,  &c. 

IV.  HIGHER  ARITHMETIC;    or,  the.  Science  and  Applica- 
tion of  Numbers  ;  —  For  advanced  classes.     This  work  is  complete 


in  itself^  commencing  with^the  fundamental  rules,  and  extending  to 

*  ^          '     v 
KEY    TO  .HIGHER    ARITHMETIC  ;—  Containing  the 


^ 
highest  deparflnent  o 


answers^tb  all  the  examples,  with  many  suggestions,  &c. 
t    VI.    ELEMENTS  OF  ALGEBRA  >*JBeinffa  SjchooAdkion 
of  Day's  «^ge  AMgebra/>,T.his'  wo?k  ^§  designed  to  be  a  ItMd  and 

"  'easy  transition  frq/n  the  study  of  Arithmetic  ro^the  higher  branches 
.of  Mathtjiiatics.  The  number  of  examples  is  much  increased  ;  and 
Jjj,e  worHfcis  every  way  adapted  to  Schools  and  Academies. 

\  VII.  KEY  TO  ELEMENTS  OF  ALGEBRA  ;—  Containing 
the  answers,  the  solution  of  the  more  difficult  problems,  &c. 

VIII.  ELEMENTS    OF    GEOMETRY;  —  Being  our  abridg- 
ment of  Legendre's    Geometry;  with   practical    notes   and  illus- 
trations. 

IX.  ELEMENTS     OF     TRIGONOMETRY,    MENSURA- 
TION. AND  LOGARITHMS. 

X.  ELEMENTS  OF   SURVEYING  ;-  Adapted  both  to  the 
wants  of   the  learner  and  the  practical  Surveyor.       (Published 
soon.) 

Entered  according  to  Act  of  Congress,  in  the  year  1845, 

BY  JEREMIAH  DAY  and  JAMES  B.  Tuoneaw, 
in  the  Clerk's  Office  of  the  District  Court  of  Connecticut 


PREFACE. 

— — — 

IT  has  been  well  said,  that  "whoever  shortens  the 
road  to  knowledge,  lengthens  life."  The  value  of  a 
knowledge  of  Arithmetic  is  too  generally  appreciated  to 
require  comment.  When  properly  studied,  two  impor- 
tant ends  are  attained,  viz :  discipline  of  mind,  and  fa- 
cility in  the  application  of  numbers  to  business  calcula- 
tions. Neither  of  these  results  can  be  secured,  unless 
the  pupil  thoroughly  understands  the  principle  of  every 
operation  he  performs.  There  is -no  uncertainty  in  the 
conclusions  of  mathematics;  there  should  be  no  guess- 
work in  its  operations.  What  then  is  the  cause  of  so 
much  groping  and  fruitless  effort  in  this  department  of  ed- 
ucation. Why  this  aimless,  mechanical  "  ciphering,"  that 
is  so  prevalent  in  our  schools  ? 

The  present  work  was  undertaken,  and  is  now  offered 
to  the  public,  with  the  hope  of  contributing  something 
toward  the  removal  of  these  inveterate  evils.  Its  plan  is 
the  following : 

1.  To  lead  the  pupil  to  a  knowledge  of  each  rule  by 
induction ;  that  is,  by  the  examination  and  solution  of  a 
large  number  of  practical  examples  which  involve  the 
principles  of  the  rule. 

2.  The  operation  is  then  defined,  each  principle  is  ana- 
lyzed separately,  and  illustrated  by  other  examples. 

3.  The  general  rule  is  now  deduced,  and  put  in  its 
proper  place,  both  for  convenient  reference  and  review ; 
thus  combining  the  inductive  and  synthetic  modes  of  in- 
struction. 

4.  The  general  rule  is  followed  by  copious  examples  for 
practice,  which  are  drawn  from  the  various  departments 
of  business,  and  are  calculated  both  to  call  into  exercise 
the  different  principles  of  the  rule,  and  to  prepare  th« 
learner  for  the  active  duties  of  life. 


IT  PREFACE. 

It  is  believed  that  much  of  this  guess-work  in  " 
ing,"  and  its  concomitant  habits  of  listlessness  and 
cuity  of  .mind,  have  arisen  from  the  use,  at  first,  of  abstrad 
numbers  and  intricate  questions,  requiring  combinations 
above  the  capacity  of  children.  Taking  his  slate  and  pen- 
cil, the  pupil  sits  down  to  the  solution  of  his  problem,  but 
soon  finds  himself  involved  in  an  impenetrable  maze.  He 
anxiously  asks  for  light,  and  is  directed  "  to  learn  the  rule." 
He  does  it  to  the  letter,  but  his  mind  is  still  in  the  dark 
By  puzzling  and  repeated  trials,  he  perhaps  finds  that 
certain  multiplications  and~ divisions  produce  the  answer 
in  the  book  ;  but  as  to  the  reasons  of  the  process,  he  is  to- 
tally ignorant.  To  require  a  pupil  to  learn  and  understand 
the  rule,  before  he  is  permitted  to  see  its  principles  illus- 
trated by  simple  practical  examples,  places  him  in  the 
condition  of  the  boy,  whose  mother  charged  him  never  to 
go  into  the  water  till  he  had  learned  to  swim. 

These  embarrassments  are  believed  to  be  unnecessary, 
and  are  attempted  to  be  removed  in  the  following  manner : 

1.  The  examples  at  the  commencement  of  each  rule 
are  all  practical,  and  are  adapted  to  illustrate  the  particular 
principle  under  consideration.     Every  teacher  can  bear 
testimony,  that  children  reason  upon  practical  questions 
with  far  greater  facility  and  accuracy  than  they  do  upon 
abstract  numbers. 

2.  The  numbers  contained  in  the  examples  are  at  first 
small,  so  that  the  learner  can  solve  the  question  mental 
ly,  and  understand  the  reason  of  each  step  in  the  opera- 
tion. 

3.  As  the  pupil  becomes  familiar  with  the  more  simple 
combinations,  the   numbers  gradually  increase,  till  the 
slate  becomes  necessary  for  the  solution,  and  its  proper 
use  is  then  explained. 

4.  Frequent  mental  exercises  are  interwoven  with  ex- 
ercises upon  the  slate,  for  the  purpose  of  strengthening  the 
habit  of  analyzing  and  reasoning,&n&  thus  enable  the  learn- 
er to  comprehend  and  solve  the  more  intricate  problems. 

5.  In  the  arrangement  of  subjects  it  has  been  a  cardi- 
nal point  to  follow  the  natural  order  of  the  science.     No 
principle  is  used  in  the  explanation  of  anpther,  until  it 


PREFACE. 


has  itself  been  demonstrated  or  explained.  Common  frac- 
tions, therefore,  are  placed  immediately  after  division,  for 
two  reasons.  First,  they  arise,  from  division,  and  are 
inseparably  connected  with  it.  Second,  in  Reduction, 
Compound  Addition,  &c.  it  is  frequently  necessary  to 
use  fractions ;  consequently  fractions  must  be  understood, 
before  it  is  possible  to  understand  the  Compound  rules. 

For  the  same  reason,  Federal  Money,  which  is  based 
upon  the  decimal  notation,  is  placed  after  Decimal  Frac- 
tions. Interest,  Insurance,  Commission,  Stocks,  Duties, 
&c.,  are  also  placed  after  Percentage,  upon  whose  prin- 
ciples they  are  based. 

6.  In  preparing  the  tables  of  Weights  and  Measures, 
particular  pains  have  been  taken  to  ascertain  those  that 
are  in  present  use  in  our  country,  and  to  give  the  If  gal 
standard  of  each,  as  adopted  by  the  General  Government.* 
It  is  well  known  that  a  great  difference  of  weights  and 
measures  formerly  existed  in  different  parts  of  the  coun- 
try.    More  than  ten  years  have  elapsed  since  the  Gov 
ernment   wisely   undertook  to   remedy  these  evils,  by 
adopting    uniform  standards  for    the  custom-houses    and 
other  purposes  ;  and  yet  not  a  single  author  of  arithme- 
tic, so  far  as  we  know,  has  given  these  standards  to  the 
public. 

7.  The  subject  of  Analysis  is  deemed  so  essential  to  a 
thorough  knowledge  of  arithmetic  and  to  business  calcu- 
lations, that  a  whole  section  is   devoted   to  its  develop- 
ment and  application.      The   principles   of   Cancelation 
have  been  illustrated,  and  its  most  important  applications 
pointed  out,  in  their  proper  places.     The   Square  and 


*  In  the  year  1836,  Congress  directed  the  Secretary  of  the  Treas- 
ury to  cause  to  be  delivered  to  the  Governor  of  each  State  in  the 
Union,  or  to  such  person  as  he  should  appoint,  a  complete  set  of  all 
the  Weights  and  Measures  adopted  as  standards,  for  the  use  of  the 
States  respectively ;  to  the  end  that  a  uniform  standard  of  Weights 
and  Measures  may  be  established  throughout  the  United  States. 
Most  of  the  States  have  already  received  them ;  and  may  we  not 
hope  that  every  member  of  this  great  Union  will  promptly  and  cor- 
dially unite  in  the  accomplishment  of  an  object  so  conducive  botK  *« 
ttjdividual  and  publir ;  good. 


VI  PREFACE. 

Cube  Roots  are  illustrated  by  geometrical  figures  and  cu- 
bical blocks. 

Such  is  a  brief  outline  of  the  present  work.  It  is  not 
designed  to  be  a  book  of  puzzles,  or  mathematical  anom- 
alies ;  but  to  present  the  elements  of  practical  arithmetic 
in  a  lucid  and  systematic  manner.  It  embraces,  in  a  word, 
all  'he  principles  and  rules  which  the  business  man  ever 
has  occasion  to  use,  and  is  particularly  adapted  to  pre- 
cede the  study  of  Algebra  and  the  higher  branches  of 
mathematics. 

With  what  success  the  plan  has  been  executed  re- 
mains for  teachers  and  practical  educators  to  decide.  If 
it  should  be  found  to  shorten  the  road  to  a  thorough  knovvL 
edge  of  arithmetic  in  any  degree,  its  highest  aims  will  be 
accomplished. 

J.  B.  THOMSON, 
New  Haven,  Oct.  3,  1845. 


SUGGESTIONS 

ON   THE 

MODE  OF  TEACHING  ARITHMETIC. 


I.  QUALIFICATIONS. — The  chief  qualifications  requisite  in  teaching 
Arithmetic,  as  well  as  other  branches  are  the  following : 

1.  A  thorough  knowledge  of  the  subject. 

2.  A  love  for  the  employment. 

3.  An  aptitude  to  teach.     These  are  indispensable  to  success. 

II.  CLASSIFICATION. — Arithmetic,  as  well  as  reading,  grammar,  &c., 
bhould  be  taught  in  classes. 

1.  This  method  saves  much  time,  and  thus  enables  the  teacher  to 
devote  more  attention  to  oral  illustrations. 

"2.  The  action  of  mind  upon  mind,  is  a  powerful  stimulant  to  exer- 
tion, and  can  not  fail  to  create  a  zest  for  the  study. 

3.  The  mode  of  analyzing  and  reasoning  of  one  scholar,  will  often 
suggest  new  ideas  to  the  others  in  the  cla-ns. 

4.  In  the  classification,  those  should  be  put  together  who  possess  as 
nearly  equal  capacities  and  attainments  as  possible.     If  any  of  the 
class  leurn  quicker  than  others,  they  should  be  allowed  to  take  up  an 
extra  study,  or  be  furnished  with  additional  examples  to  solve,  so 
that  the  whole  class  may  advance  together. 

5.  The  number  in  a  class,  if  practicable,  should  not  be  less  than 
six,  nor  over  twelve  or  fifteen.     If  the  number  is  less,  the  recitation 
is  apt  to  be  deficient  in  animation  ;  if  greater,  the  turn  to  recite  does 
not  come  round  sufficiently  often  to  keep  up  the  interest. 

III.  APPARATUS. — The  Black-board  and  Numerical  frame  are  as 
indispensable  to  the  teacher,  as  tables  and  cutlery  are  to  the  house- 
keeper.    Not  a  recitation  passes  without  use  for  the  black-board.     If 
a  principle  is  to  be  demonstrated  or  an  operation  explained,  it  should 
be  done  upon  the  black-board,  so  that  all  may  see  and  understand  it 
at  once. 

To  illustrate  the  increase  of  numbers,  the  process  of  adding,  sub- 
tracting, multiplying,  dividing,  &c.,  the  Numerical  Frame  furnishes 
one  of  the  most  simple  and  convenient  methods  ever  invented.* 

IV.  RECITATIONS. — The  Jirst  object  in  a  recitation,  is  to  secure 
the  attention  of  the  class.     This  is  done  chiefly  by  throwing  life  and 
variety  into  the  exercise.     Children  loathe  dullness,  while  animation 
and  variety  are  their  delight. 

2.  The  teacher  should  not  be  too  much  confined  to  his  text-book, 
nor  depend  upon  it  wholly  for  illustrations. 

*  Every  one  who  ciphers,  will  of  course  have  a  slate.  Indeed,  it  is  desira- 
ble that  every  scholar  in  school,  even  to  the  very  youngest,  should  be  fur- 
nished with  a  small  slate,  so  that  when  the  little  fellows  have  learned  their 
lessons,  they  may  busy  themselves  in  writing  and  drawing  various  familiar 
objects.  Idleness  in  school  is  the  parent  of  mischief,  and  employment  is  the  best 
antidote  against  disobedience. 

Geometrical  diagrams  and  solids  are  also  highly  useful  in  illustrating  manf 
points  in  arithmetic,  and  no  school  should  hp  without  them 


VI  PREFACE. 

Cube  Roots  are  illustrated  by  geometrical  figures  and  cu- 
bical blocks. 

Such  is  a  brief  outline  of  the  present  work.  It  is  not 
designed  to  be  a  book  of  puzzles,  or  mathematical  anom- 
alies ;  but  to  present  the  elements  of  practical  arithmetic 
in  a  lucid  and  systematic  manner.  It  embraces,  in  a  word, 
all  he  principles  and  rules  which  the  business  man  ever 
has  occasion  to  use,  and  is  particularly  adapted  to  pre- 
cede the  study  of  Algebra  and  the  higher  branches  of 
mathematics. 

With  what  success  the  plan  has  been  executed  re- 
mains for  teachers  and  practical  educators  to  decide.  If 
it  should  be  found  to  shorten  the  road  to  a  thorough  knovvL 
edge  of  arithmetic  in  any  degree,  its  highest  aims  will  be 
accomplished. 

J.  B.  THOMSON, 
New  Haven,  Oct.  3,  1845. 


SUGGESTIONS 

ON   THE 

MODE  OF  TEACHING  ARITHMETIC. 


I.  QUALIFICATIONS. — The  chief  qualifications  requisite  in  teaching 
Arithmetic,  as  well  as  other  branches  are  the  following : 

1.  A  thorough  knowledge  of  the  subject. 

2.  A  love  for  the  employment. 

3.  An  aptitude  to  teach.     These  are  indispensable  to  success. 

II.  CLASSF ^CATION. — Arithmetic,  as  well  as  reading,  grammar,  &c., 
should  be  taught  in  classes. 

1.  This  method  saves  much  time,  and  thus  enables  the  teacher  to 
devote  more  attention  to  oral  illustrations. 

2.  The  action  of  mind  upon  mind,  is  a  powerful  stimulant  to  exer- 
tion, and  can  not  fail  to  create  a  zest  for  the  study. 

3.  The  mode  of  analyzing  and  reasoning  of  one  scholar,  will  often 
suggest  new  ideas  to  the  others  in  the  cla-ss. 

4.  In  the  classification,  those  should  be  put  together  who  possess  as 
nearly  equal  capacities  and  attainments  as  possible.     If  any  of  the 
class  leurn  quicker  than  others,  they  should  be  allowed  to  take  up  an 
extra  study,  or  be  furnished  with  additional  examples  to  solve,  so 
that  the  whole  class  may  advance  together. 

5.  The  number  in  a  class,  if  practicable,  should  not  be  less  than 
six,  nor  over  twelve  or  fifteen.     If  the  number  is  less,  the  recitation 
is  apt  to  be  deficient  in  animation  ;  if  greater,  the  turn  to  recite  does 
not  come  round  sufficiently  often  to  keep  up  the  interest. 

III.  APPARATUS. — The  Black-board  and  Numerical  Frame  are  as 
indispensable  to  the  teacher,  as  tables  and  cutlery  are  to  the  house- 
keeper.    Not  a  recitation  passes  without  use  for  the  black-board.     If 
a  principle  is  to  be  demonstrated  or  an  operation  explained,  it  should 
be  done  upon  the  black-board,  so  that  ail  may  see  and  understand  it 
at  once. 

To  illustrate  the  increase  of  numbers,  the  process  of  adding,  sub- 
tracting, multiplying,  dividing,  &c.,  the  Numerical  Frame  furnishes 
one  of  the  most  simple  and  convenient  methods  ever  invented.* 

IV.  RECITATIONS. — The  first  object  in  a  recitation,  is  to  secure 
the  attention  of  the  class.     This  is  done  chiefly  by  throwing  life  and 
variety  into  the  exercise.     Children  loathe  dullness,  while  animation 
and  variety  are  their  delight. 

2.  The  teacher  should  not  be  too  much  confined  to  his  text-book, 
nor  depend  upon  it  wholly  for  illustrations. 

*  Every  one  who  ciphers,  will  of  course  have  a  slate.  Indeed,  it  is  desira- 
ble that  every  scholar  in  school,  even  to  the  very  youngest,  should  be  fur- 
nished with  a  small  slate,  so  that  when  the  little  fellows  have  learned  their 
lessons,  they  may  busy  themselves  in  writing  and  drawing  various  familiar 
objects.  Illencss  in  school  is  the  parent  of  mischief,  and  employment  is  the  best 
antidote  against  disobedience. 

Geometrical  diagrams  and  solids  are  also  highly  useful  in  illustrating  many 
points  in  arithmetic,  and  no  school  should  Iw  without  them 


Vlll  SUGGEST  JONS. 

3.  Every  example  should  be  analyzed,  the  "  why  and  wherefcre ' 
of  every  step  in  the  solution  should  be  required,  till  each  member  ot 
the  class  becomes  perfectly  familiar  with  the  process  of  reasoning  and 
analysis. 

4.  To  ascertain  whether  each  pupil  has  the  right  answer  to  all  the 
examples,  it  is  an  excellent  method  to  name  a  question,  then  call 
upon  some  one  to  give  the  answer,  and  before  deciding  whether  it  ia 
right  or   wrong,  ask  how  many  in  the  class  agree  with  it.     The  an- 
/wer  they  give  by  raising  their  hand,  will  show  at  once  how  many 
are  right.     The  explanation  of  the  process  may  now  be  made. 

Another  method  is  to  let  the  class  exchange  slates  with  each  other, 
and  when  an  answer  is  decided  to  be  right  or  wrong,  let  every  one 
mark  it  accordingly.  After  the  slates  are  returned  to  their  owners, 
each  one  will  correct  his  errors. 

V.  THOROUGHNESS. — The  motto  of  every  teacher  should  be  thor- 
oughness.    Without  it,  the  great  ends  of  the  study  are  defeated. 

1.  In  securing  this  object,  much  advantage  is  derived  from  fre- 
qutnt  reviews. 

2.  Not  a  recitation  should  pass  without  practical  exercises  upon 
the  black-board  or  slates,  besides  the  lesson  assigned. 

3.  After  the  class  have  solved  the  examples  under  a  rule,  each  one 
should  be  required  to  give  an  accurate  account  of  its  principles  with 
the  reason  for  each  step,  either  in  his  own  language  or  that  of  the 
author. 

4.  Mental  Exercises  in  arithmetic,  either  by  classes  or  the  whole 
school  together,  are  exceedingly  useful  in  making  ready  and  accurate 
arithmeticians,  and  should  be  frequently  practised. 

VI.  SELF-RELIANCE. — The  habit  of  self-reliance  in  study,  is  confess- 
edly invaluable.     Its  power  is  proverbial ;  I  had  almost  said,  omnipo- 
tent.    "  Where  there  is  a  will,  there  is  a  way." 

1.  To  acquire  this  habit,  the  pupil,  like  a  child  learning  to  walk, 
must  be  taught  to  depend  upon  himself.     Hence, 

2.  When  assistance  in  solving  an  example  is  required,  it  should  be 
given  indirectly ;  not  by  taking  the  slate  and  performing  the  exam- 
ple for  him,  but  by  explaining  the  meaning  of  the  question,  or  illus- 
trating the  principle  on  which  the  operation  depends,  by  supposing  a 
more  familiar  case.     Thus  the  pupil  will  be  able  to  solve  the  question 
himself,  and  his  eye  will  sparkle  with  the  consciousness  of  victory. 

3.  He  must  learn  to  perform  examples  independent  of  the  answer, 
without  seeing  or  knowing  what  it  is.     Without  this  attainment  the 
pupil  receives  but  little  or  no  discipline  from  the  study,  and  is  unfit  to 
be  trusted  with  business  calculations.     What  though  he  comes  to  the 
recitation  with  an  occasional  wrong  answer ;  it  were  better  to  solve  ono 
question  understandingly  and  alone,  than  to  copy  a  score  of  answers 
from  the  book.     What  would  the  study  of  mental  arithmetic  be  worth, 
if  the  pupil  had  the  answers  before  him  1   What  is  a  young  man  good 
for  in  the  counting-room,  who  has  never  learned  to  perform  arithmeti- 
cal operations  alone,  but  is  obliged  to  look  to  the  answer  to  know  what 
figure  to  place  in  the  quotient,  or  what  number  to  place  for  the  third 
term  in  proportion,  as  is  too  often  the  case  in  school  ciphering  7 


CONTENTS. 


SECTION  I. 

Page- 

Suggestions  on  the  mode  of  teaching  Arithmetic,    .        .        , 

Numbers  illustrated  and  defined, 13 

Notation,         ..........  14 

Roman  Notation,     .........  14 

Aral)i"  Notation,      .........  15 

Numeration,  .........  19 

Exercises  in  Numeration,        .......  2t 

Exercises  in  Notation,  .......  21 

SECTION  II. 

ADDITION,  Mental  Exercises, 23 

Addition  Table, 24 

Exercises  for  the  Slate, 28 

Illustration  of  the  principle  of  carrying,  32 

Proof  of  Addition, 34 

General  Rule, 34 

Examples  for  practice, 35 

SECTION  III. 

SUBTRACTION,  Mental  Exercises, .40 

Subtraction  Table, 41 

Exercises  for  the  Slate, 44 

Illustration  of  the  principle  of  borrowing,         ....  46 

Proof  of  Subtraction, 49 

General  Rule, 49 

Examples  for  practice,. 50 

SECTION  IV. 

MULTIPLICATION,  Mental  Exercises, 54 

Exercises  for  the  Slate,  ...  ...  GO 

Illustration  of  carrying  in  Multiplication,        ....  62 

General  Rule, 66 

Examples  for  practice, 66 

Contractions  in  Multiplication, 68 


X  CONTENTS. 

SECTION  V. 

DIVISION,  Mental  Exercises, 73 

Exercises  for  the  Slate, 78 

Short  Division, 82 

Long  Division, 83 

Contractions  in  Division, 87 

General  principles  in  Division,      .......  90 

CANCELATION  illustrated, 93 

Greatest  Common  Divisor,            95 

Least  Common  Multiple, 97 

SECTION  VI. 

FRACTIONS,  Mental  Exercises,       .        .        .        .        .         .  100 

Reduction  of  Fractions, 108 

Cancelation  applied  to  reducing  Compound  Fractious,          :  112 

Common  Denominator, 113 

Least  Common  Denominator,        .        .        ..        .        ,        .  114 

Addition  of  Fractions, 115 

Subtraction  of  Fractions, 117 

Multiplication  of  Fractions, 120 

Cancelation  applied  to  Multiplication  of  Fractions,        .        .  125 

Division  of  Fractions,            128 

Cancelation  applied  to  Division  of  Fractions,         .         .         .  132 

Cancelation  applied  10  Multiplication  of  Complex  Fractions,  134 


SECTION  VII. 

COMPOUND  NUMBERS, 136 

Tables  in  Compound  Numbers, .  137 

The  standard  unit  of  Weight  of  the  United  States,         .         .  138 
The  Avoirdupois  Pound  of  the  United  States  and  Great  Britain,  139 

The  standard  unit  of  Length  of  the  United  States,       .         .  140 

The  standard  unit  of  Liquid  Measure  of  the  United  States,    .  144 

The  standard  unit  of  Dry  Measure  of  the  United  States,       .  146 

REDUCTION, ]49 

To  find  the  area  of  surfaces, 154 

To  find  the  solidity  of  boxes,  wood,  &c.,       ....  155 

Compound  numbers  reduced  to  fractions,      .         .         .         .  157 

Fractional  Compound  numbers  reduced  to  whole  numbers,  .  159 

Compound  Addition, 160 

Compound  Subtraction, 163 

To  find  the  difference  between  two  Dates,    ....  165 

Compound  Multiplication, 166 

Compound  Division, 169 


©OflTENTS.  XI 

SECTION  VIII. 

DECIMAL  FRACTIONS,            . 171 

Exercises  in  reading  and  writing  Decimals,           .        .        .  174 

Addition  of  Decimals,             .         .         .        .        .        .        .  175 

Subtraction  of  Decimals, 177 

Multiplication  of  Decimals, 179 

Division  of  Decimals,             .                 181 

Reduction  of  Decimals, 184 

Decimals  reduced  to  Common  Fractions,      . 

Common  Fractions  reduced  to  Decimals,       '.        .        .        .  185 

Circulating  Decimals,             186 

Compound  numbers  reduced  to  Decimals  of  higher  denom., 
Decimal  Compound  numbers  reduced  to  whole  numbers, 

FEDERAL  MONEY, 

Reduction  of  Federal  Money, 191 

Addition  of  Federal  Money,           .         •        .         .        .        .  193 

Subtraction  of  Federal  Money, 194 

Multiplication  of  Federal  Money,          .        .        .        .        .  195 
When  the  price  of  one  article,  &c.,  is  given,  to  find  the  cost  of 

any  number  of  articles, 

Division  of  Federal  Money 199 

When  the  number  of  articles,  &c.,  and  the  cost  of  the  whole  are 

given,  to  find  the  price  of  one,            .  199 
When  the  price  of  one  article  and  the  cost  of  the  whole  are  given, 

to  find  the  number  of  articles,            200 

Applications  of  Federal  Money  to  Bills,  &c.,        .        .        .  202 

SECTION  IX. 

PERCENTAGE, 204 

Application  of  Percentage,            .  210 

Commission,  Brokerage  and  Stocks, 211 

INTEREST,              214 

General  method  for  computing  Interest,         ....  220 

Second  method        "                                      ....  223 

Partial  payments,  Rule  adopted  by  the  United  States,            .  227 

Connecticut  and  Vermont  Rules, 229 

Problems  in  Interest,             231 

Compound  Interest, 236 

DISCOUNT,            239 

Bank  Discount,             242 

Insurance,              244 

Profit  and  Loss,             247 

DUTIES, 256 

Specific  Duties,             • 257 

Ad  valorem  Duties, 258 

Taxes,  assessment  of, .  260 


Xll 


CONTENTS. 


SECTION  X. 

PROPERTIES  OP  NUMBERS.  

Proof  of  Multiplication  and  Division  from  the  property  of  9, 

Axioms, 

Deductions  from  the  Fundame»tal  Rules, 

SECTION  XI. 


Pa** 

264 
267 
268 
269 


ANALYSIS, 
Analyticsolut 
Ditto 
Ditto 
Ditto 
Ditto 
Ditto 
Ditto 
Ditto 
Ditto 

ons  of  qu 

c 
I 

potions  in  Simple  Proportion, 
Barter, 
Practice, 
Partnership, 
Bankruptcy, 
General  Average, 
Alligation, 
Comp'd  Proportion, 
Position, 

273 
Ex.  1-50,        276 

Ex.  51-60,     28-2 
Ex.  61  -75,     283 
Ex.  76-82,     284 
Ex.  83-88,     285 
Ex.  89-9  1,     286 
Ex.  9-2-1  00,   287 
Ex.  101-106,  288 
Ex.  107-120.  289 

SECTION  XII. 

RATIO, 

PROPORTION,         .        . 

Simple  Proportion,  or  Rule  of  Three, 

Compound  Proportion,  or  Double  Rule  of  Three, 


290 
294 
294 
307 


SECTION   XIII. 


Duodecimals, 

Multiplication  of  Duodecimals, 


SECTION   XIV. 


Involution,  

Evolution,  

Square  Root, 

Applications  of  Square  Root, 

Cube  Root, 

Demonstration  of  Cube  Root  by  Cubical  Blocks, 


SECTION   XV. 


Equation  of  Payments, 
Partnership  or  Fellowship, 
Exchange  of  Currencies, 

MENSURATION,       . 

Miscellaneous  Examples, 


306 
307 


310 
313 
317 
322 
324 
325 


328 
331 
334 
338 
344 


ARITHMETIC, 

SECTION    I. 

NOTATION  AND   NUMERATION.    ; 

ART.  1  •  Any  single  thing,  as  a  peach,  a  rose,  a  book, 
is  called  a  unit,  or  one  /  if  another  single  thing  is  put 
with  it,  the  collection  is  called  two  ;  if  another  still,  it  is 
called  three  •  if  another,  four  •  if  another,  Jive,  &c. 

The  terms,  one,  two,  three,  &c.>  by  which  we  express 
how  many  single  things  or  units  are  under  consideration, 
are  the  names  of  numbers.  Hence, 

2.  NUMBER  signifies  a  unit,  or  a  collection  of  units. 

OES.  Numbers  have  various  properties  and  relations,  and  are  ap- 
plied to  various  calculations  in  the  practical  concerns  of  life.  These 
properties  and  applications  are  formed  into  a  system,  called  Arith- 
metic. Hence, 

3*  ARITHMETIC  is  t/ie  science  of  numbers. 

Numbers  are  expressed  by  words,  by  letters,  and  by 
figures. 

Note. — The  questions  on  the  observations  may  be  omitted,  by  be- 
ginners, till  review,  if  deemed  advisable  by  the  Teacher. 

QUEST. — 1.  What  is  a  single  thing  called  ?  If  another  ^s  put  with 
it,  what  is  the  collection  called  ?  If  another,  what  ?  What  are  the  terms 
one,  two,  three,  &c.  ?  2.  What  does  number  signify  ?  Obs.  To  what 
are  numbers  applied  ?  3.  What  is  Arithmetic  ?  Ho\\  are  numbers  ex- 
pressed ? 


14 


NOTATION. 


[SECT.  I. 


NOTATION. 

4»  The  art  of  expressing  numbers  by  letters  or  figures, 
is  called  NOTATION.  There  are  two  methods  of  notation 
in  use,  the  Roman  and  the  Arabic. 

5  •  The  Roman  method  employs  seven  capital  letters, 
viz :  I,  V5  X,  L,  C,  D,  M.  When  standing  alone,  the  letter 
I  denotes  one',  V,five;  X,  fe#;  L, fifty  ;  C,  one  hundred; 
D,  five  hundred ;  M,  one  thousand.  To  express  the  in- 
tervening numbers  from  one  to  a  thousand,  or  any  number 
larger  than  a  thousand,  we  resort  to  repetitions  and  various 
combinations  of  these  letters.  The  method  of  doing  this 
will  be  easily  learned  from  the  following 

TABLE. 


I        denotes  one. 

XXX  denote  thirty. 

II            "      two. 

XL          "      forty. 

Ill            <       three. 

L             "      fifty. 

IV            '      four. 

LX          "      sixty. 

V              '       five. 

LXX       "      seventy. 

VI            (      six. 

LXXX  "      eighty. 

VII           '       seven. 

XC          "      ninety. 

VIII        "       eight. 

C              "      one  hundred. 

IX 

'       nine. 

CI                   one  hundred  and  ona 

X 

'       ten. 

CX           '      one  hundred  and  ten, 

XI 

'       eleven. 

CC            '      two  hundred. 

XII 

xln 

'       twelve, 
thirteen. 

CCC         «      three  hundred. 
CCCC      '      four  hundred. 

XIV 

fourteen. 

D             "      five  hundred. 

XV 

'       fifteen. 

DC                 six  hundred. 

XVI 

'       sixteen. 

DCC               seven  hundred. 

XVII 

'       seventeen. 

DCCC            eight  hundred. 

XVIII 

;       eighteen. 

DCCCC   '      nine  hundred. 

XIX 

'       nineteen. 

M              '      one  thousand. 

XX 

'       twenty. 

MM          '      two  thousand. 

XXI 
XXII 

'       twenty-one. 
'       twenty-two,  &c. 

MDCCCXLV,  one  thousand  eight 
hundred  and  forty-five. 

QUEST.-4.  What  is  notation  ?  How  many  methods  are  there  in  use  ? 
What  are  they  t  5.  What  does  the  Roman  method  employ  ?  Wha . 
does  each  of  these  letters  denote  when  standing  alone  ?  How  are  thti 
intervening  numbers  from  one  to  a  thousand  expressed?  How  denote 
Two?  Four?  Six?  Eight?  Nine?  Fourteen  ?  Sixteen?  Nineteen? 
Twenty-four  ?  Twenty-eight  ?  What  does  XL  denote  ?  LX  ?  XC  ?  CX  ? 

N.  B.  Questions  on  this  table  should  be  varied,  and  continued  by  the 
Readier  til!  the  etass  becomes  perfectly  familiar  witU  it. 


ARTS.  4 — 7.]  K^TATION.  15 

OBS.  1.  The  learner  will  perceive  from  the  Table  above,  that  every 
time  a  letter  is  repeated,  its  value  is  repeated.  Thus  I,  standing  alone, 
denotes  one ;  II,  two  ones  or  two,  &c.  So  X  denotes  ten;  XX, 
twenty,  &c. 

2.  When  two  letters  of  different  value  are  joined  together,  if  the 
less  is  placed  before  the  greater,  the  value  of  the  greater  is  dimin- 
ished; if  placed  after  the  greater,  the  value  of  the  greater  is  increased, 
Thus,  V  denotes  fiv«  ;  but  IV  denotes  only  four ;  and  VI,  six.     So  X 
denotes  ten ;  IX,  nine  ;  XI,  eleven. 

3.  A  line  or  bar  ( — )  placed  over  a_letter,  increases  its  value  a 
thousand  times.     Thus,  V  denotes  five,  V  denotes  five  thousand ;  X, 
ten ;  X,  ten  thousand. 

4.  This  method  of  expressing  numbers  was  invented  by  the  Ro- 
mans ;  hence  it  is  called  the  Roman  Notation.     It  is  now  seldom  used, 
except  to  denote  chapters,  sections,  and  other  divisions  of  books  and 
discourses. 

6.  The  common  method  of  expressing  numbers  is  by 
the  Arabic  Notation.  The  Arabic  method  employs  the 
following  ten  characters  or  figures,  viz  : 

1234567          8          90 
one,  two,  three,  four,  five,  six,  seven,  eight,  nine,  zero. 

The  first  nine  are  called  significant  figures,  because 
each  one  always  has  a  value,  or  denotes  some  number. 
They  are  also  called  digits,  from  the  Latin  word  digitus, 
tvhich  signifies  a  finger. 

The  last  one  is  called  a  cipher,  or  naught,  because  when 
standing  alone  it  has  no  value,  or  signifies  nothing. 

OBS.  It  must  not  be  inferred,  however,  that  the  cipher  is  useless ;  for 
when  placed  on  the  right  of  any  of  the  significant  figures,  it  increases 
I  heir  value.  It  may  therefore  be  regarded  as  an  auxiliary  digit,  whoso 
office,  it  will  be  seen  hereafter,  is  as  important  as  that  of  any  other 
figure  in  the  system. 

Nrte. — The  pupil  must  be  able  to  distinguish  and  to  write  these 
characters,  before  he  can  make  any  progress  in  Arithmetic. 

7  •  It  will  be  seen  that  nine  is  the  greatest  number  that 


QUEST. — Ols.  What  is  the  effect  of  repeating  a  letter?  If  a  let:  . 
is  placed  before  another  of  greater  value,  what  is  the  effect  ?  If  placed 
after,  whai  ?  When  a  letter  has  a  line  placed  over  it,  how  is  its  valu-- 
affected  ?  Why  is  this  method  of  notation  called  Roman  ?  To  what 
use  is  it  chiefly  applied  ?  6.  How  are  numbers  commonly  expressed  ? 
How  many  characters  does  this  method  employ?  What  are  their 
names  ?  What  are  the  first  nine  called  ?  Why  ?  What  else  are  they 
called?  What  is  the  last  one  called?  Why?  Obs.  In  the  (ipher 
useless  ?  What  may  U  be  regarded  ? 


16 


NOTATION. 


[SECT   i 


can  be  expressed  by  any  single  figure.  All  numbers  larger 
than  nine  are  expressed  by  combining  together  two  or 
more  of  the  ten  characters  just  explained.  To  express 
ten  for  example,  we  combine  the  1  and  0,  thus  10  ;  eleven 
is  expressed  by  two  Is,  thus  11;  twelve,  thus  12;  two 
tens,  or  twenty,  thus  20  ;  one  hundred,  thus  100,  &c, 

The  numbers  from  one  to  a  thousand  are  expressed  ir 
the  following  manner : 


1,  one. 

2,  two. 

3,  three. 

4,  four. 

5,  five. 

6,  six. 

7,  seven. 

8,  eight. 

9,  nine. 

10,  ten. 

11,  eleven. 

12,  twelve. 

13,  thirteen. 

14,  fourteen. 

15,  fifteen. 
16  sixteen. 

17,  seventeen. 

18,  eighteen. 

19,  nineteen. 

20,  twenty. 

21,  twenty-one,  &c. 

30,  thirty. 

31,  thirty-one,  &c. 

40,  forty. 

41,  forty-one,  &c. 

50,  fifty. 

51,  fifty-one,  &c. 
CO,  sixty. 

61,  sixty-one,  &c. 

70,  seventy. 

71,  seventy-one,  &c. 
80,  eighty. 


81,  eighty-one,  &c. 

90,  ninety. 

91,  ninety-one,  &c. 

100,  one  hundred. 

101,  one  hundred  and  one. 

102,  one  hundred  and  two. 

103,  one  hundred  and  three. 

110,  one  hundred  and  ten. 

111,  one  hundred  and  eleven. 

112,  one  hundred  and  twelve. 
120,  one  hundred  and  twenty. 
130,  one  hundred  and  thirty. 
140,  one  hundred  and  forty. 
150,  one  hundred  and  fifty. 
160,  one  hundred  and  sixty. 
170,  one  hundred  and  seventy. 
180,  one  hundred  and  eighty. 
190,  one  hundred 'and  ninety 
200,  two  hundred. 

300,  three  hundred. 
400,  four  hundred. 
500,  five  hundred. 
600,  six  hundred. 
700,  seven  hundred. 
800,  eight  hundred. 
900,  nine  hundred. 

990,  nine  hundred  and  ninety. 

991,  nine  hundred  and  ninety- one 

992,  nine  hundred  and  ninety-two 

998,  nine  hundred  &  ninety-eight 

999,  nine  hundred  &  ninety-nine. 

1000,  one  thousand. 


QUEST. — 7.  What  is  the  greatest  number  that  can  be  expressed  by 
one  figure  ?  How  are  larger  numbers  expressed  ?  How  express  ten  t 
Eleven  ?  Twelve  ?  Twenty  \  What  is  the  greatest  number  that  car. 
be  expressed  by  two  figures  ?  How  express  a  hundred  ?  One  hundred 
and  ten  ?  One  hundred  and  forty-five  ?  Five  hundred  and  sixty-eight  1 
What  is  the  greatest  number  that  can  be  expressed  by  three  figures '. 
How  express  a  thousand  ? 


ART.  8.J  NOTATION.  17 

Note. — Questions  on  the  foregoing  table  should  be  continued  till  the 
rlaw  becomes  familiar  with  the  mode  of  expressing  any  number  from 
1  tc  1000.  They  may  be  answered  orally ;  but  the  best  way  is  to  let 
the  pupil  write  the  figures  denoting  the  number  upon  the  black- 
board, and  at  the  same  time  pronounce  the  answer  audibly. 

OBS.  1.  The  terms  thirteen,  fourteen,  fifteen,  &c.,  are  obviously 
derived  from  three  and  ten,  four  and  ten,  five  and  ten,  which  by 
contraction  become  thirteen,  fourteen,  fifteen,  &c.,  and  are  therefore 
significant  of  the  numbers  which  they  denote.  The  terms  eleven 
and  twelve,  are  generally  regarded  as  primitive  words ;  at  all  events, 
there  is  no  perceptible  analogy  between  them  and  the  numbers  which 
they  represent.  Had  the  terms  onetcen  and  twoteen  been  adopted  in 
their  stead,  the  names  would  then  have  been  significant  of  the  num- 
bers one  and  ten,  two  and  ten ;  and  their  etymology  would  have  been 
similar  to  that  of  the  succeeding  terms. 

2.  The  terms  twenty,  thirty,  forty,  &c.,  were  formed  from  two 
tens,  three  tens,  four  tens,  which  were  contracted  into  twenty,  thirty 
forty,  &c. 

3.  The  terms  twenty-one,  twenty-two,  twenty-three,  &c.,  are  com- 
pounded of  twenty  and  one,  twenty  and  two,  &c.     All  the  other 
numbers  as  far  as  ninety-nine  are  formed  in  a  similar  manner. 

4.  The  terms  hundred  and  thousand  are  primitive  words,  and  bear 
no  analogy  to  the  numbers  which  they  denote.     The  numbers  be- 
tween a  hundred  and  a  thousand  are  expressed  by  a  repetition  of  the 
number  below  a  hundred.     Thus  we  say,  one  hundred  and  one,  one 
hundred  and  two,  one  hundred  and  three,  &c. 

8.  It  will  be  perceived  from  the  foregoing-  table,  that 
the  figures  standing  in  different  places  have  different  val- 
ues. Thus  the  digits,  1, 2,  3,  &c.,  standing  alone  or  in  the 
right  hand  place,  respectively  denote  units  orj)nes.  But 
when  they  stand  in  the  second  place,  they  express  tens ; 
thus  the  1  in  10, 12,  15,  &c.,  expresses  tensor  ten  ones;  that 
is,  its  value  is  ten  times  as  much  as  when  it  stands  in  the 
first  or  right  hand  place,  and  it  is  called  a  unit  of  the  sec- 
ond order.  So  the  other  digits,  2,  3,  4,  &c.,  standing 


QUEST. — Obs.  From  what  is  the  term  thirteen  formed  ?  Fourteen  ? 
Sixteen  ?  Eighteen  ?  What  is  said  of  the  terms  eleven  and  twelve  ?  How 
are  the  terms  twenty,  thirty,  &c.,  formed?  What  is  said  of  the  terms 
hundred,  and  thousand  ?  How  are  the  numbers  between  a  hundred  and 
a  thousand  expressed  ?  8.  Does  the  same  figure  always  express  the 
Fame  value  ?  What  does  each  of  the  digits,  1,  2,  3,  &c.,  denote,  when 
standing  in  the  right  hand  place  ?  What  does  the  figure  1  Jenote  when 
It  stands  in  the  second  place  1  What  is  its  value  then  ?  What  do  the 
other  figures  denote  when  standing  in  the  second  place  ? 


18  NOTATION.  [SECT.  I. 

in  the  second  place,  denote  two  tens,  three  tens,  finer 
tens,  &c. 

When  standing1  in  the  thicd  place,  they  express  hun- 
dreds: thus  the  1  in  100,  102,  123,  &c.,  denotes  a  hun- 
dred, or  ten  tens ;  that  is,  its  value  is  ten  times  as  much  as 
when  it  stands  in  the  second  place,  and  it  is  called  a  unit 
of  the  third  order.  In  like  manner,  2,  3,  4,  &c.,  standing 
hi  the  third  place,  denote  two  hundred,  three  hundred,  four 
hundred,  &c. 

When  a  digit  occupies  the  fourth  place,  it  expresses 
thousands:  thus  the  1  in  1000,  1845,  &c.,  denotes  a  thou- 
sand, or  ten  hundreds;  that  is,  its  value  is  ten  times  as 
much  as  when  it  stands  in  the  third  place,  and  it  is  called 
a  unit  of  the  fourth  order.  Thus, 

It  will  be  seen  that  ten  units  make  one  ten,  ten  tens 
make  one  hundred,  and  ten  hundreds  make  one  thou- 
sand ;  that  is,  ten  in  an  inferior  order  are  equal  to  one  in 
the  next  superior  order.  Hence,  we  may  infer  universal- 
ly, that 

9.  Numbers  increase  from  right  to  left  in  a  tenfold  ratio; 
that  is,  each  removal  of  a  figure  one  place  towards  the  left, 
increases  its  value  ten  times. 

1 0.  The  different  values  which  the  same  figures  have, 
are  called  simple  and  local  values. 

The  simple  value  of  a  figure  is  the  value  which  it  ex- 
presses when  it  stands  alone,  or  in  the  right  hand  place. 
The  simple  value  of  a  figure,  therefore,  is  the  number 
which  its  name  denotes.  (Art.  6.) 

The  local  value  of  a  figure  is  the  increased  value  which 

QUEST. — What  is  a  figure  called  when  it  occupies  the  third  place  \ 
What  is  its  value  then  \  What  is  it  called  when  in  the  fourth  place  ? 
What  is  its  value  ?  What  do  the  other  figures  denote  when  standing 
in  the  fourth  place  ?  How  many  units  are  required  to  make  one  ten  ? 
How  many  tens  make  a  hundred  ?  How  many  hundreds  make  a  thou- 
sand ?  Generally,  how  many  of  an  inferior  order  are  required  to  make 
one  of  the  next  superior  order  ?  9.  What  is  the  general  law  by  which 
numbers  increase  ?  What  is  the  effect  upon  the  value  of  a  figure  to 
remove  it  one  place  towards  the  left  ?  10.  What  are  the  different  va- 
lues of  the  same  figure  called  ?  What  is  the  simple  value  of  a  figure  \ 
What  the  local  value  ?  Upon  what  does  the  local  value  of  a  figure  de- 
pend ?  Obs.  Why  is  this  system  of  notation  called  Arabic  J  WhiU 
else  is  it  sometimes  called  ?  Why  ? 


ARTS.  9 — 12.]  NOTATION.  19 

it  expresses  by  having  other  figures  placed  on  its  right 
Hence,  the  local  value  of  a  figure  depends  on  its  locality, 
or  tbje  place  which  it  occupies  in  relation  to  other  num- 
bers (with  which  it  is  connected.  (Art.  8.) 

Osis.  1.  This  system  of  notation  is  called  Arabic,  because  it  is  sup- 
•iosed  to  have  been  invented  by  the  Arabs. 

'2.  It  is  also  called  the  decimal  system,  because  numbers  increase 
in  a  tenfold  ratio.  The  term  decimal  is  derived  from  the  Latin  word 
decem,  which  signifies  ten. 

XX.  The  art  of  reading  numbers  when  expressed  by  fig- 
ures, is  called  NUMERATION. 

The  pupil  has  already  become  acquainted  with  the 
names  of  numbers,  from  one  to  a  thousand.  He  will 
now  easily  learn  to  read  and  express  the  higher  numbers 
in  common  use,  from  the  following  scheme,  called  the 

NUMERATION  TABLE. 


<*-i     ri 
Q    S* 


568,     342, 

Period  VI.     Period  V.     Period  IV.     Period  III.    Period  II.      Period  I. 

Quadrillions.  Trillions.       Billions.        Millions.    Thousands.      Units. 

X  2«  The  different  orders  of  numbers  are  divided  into 
periods  of  three  figures  each,  beginning  at  the  right  hand. 
The  first,  which  is  occupied  by  units,  tens  and  hundreds, 

QUEST.  —  11.  What  is  numeration  ?  Repeat  the  Numeration  Table, 
beginning  at  the  right  hand.  What  is  the  first  place  on  the  right  called  l 
The  second  place  ?  The  third  ?  Fourth  ?  Fifth  ?  Sixth  ?  Seventh  ? 
Eighth  ?  Ninth  ?  Tenth,  &c.  ?  12.  How  are  the  orders  of  numbers  di- 
vided ?  What  is  the  first  period  called  ?  By  what  is  it  occupied  ? 
What  is  the  second  called  ?  By  what  occupied  ?  What  is  the  third 
called  ?  By  what  occupied  ?  What  is  the  fourth  called  ?  By  what 
occupied  *  What  is  the  fit'tL  caded  ?  By  what  occupied  1 


20  NUMERATION.  [SECT.   L 

is  called  units1  period ;  the  second  is  occupied  by  thou. 
sands,  tens  of  thousands  and  hundreds  of  thousands,  and 
is  called  thousands'  period,  &c. 

The  figures  in  the  table  are  read  thus :  Five  hundred 
and  sixty-eight  quadrillions,  three  hundred  and  forty-two 
trillions,  nine  hundred  and  seventy-five  billions,  eight  hun- 
dred and  ninety-seven  millions,  six  hundred  and  forty-five 
thousand,  four  hundred  and  thirty-two. 

1 3*  To  read  numbers  which  are  expressed  by  figures. 

Point  them  off  into  periods  of  three  figures  each  ;  then,  be- 
ginning at  the  left  hand,  read  the  figures  of  each  period  in 
the  same  manner  as  those  of  the  right  hand  period  are  read, 
and  at  the  end  of  each  period  pronounce  its  name. 

OBS.  1.  The  learner  must  be  careful,  in  pointing  of  figures,  al- 
ways to  begin  at  the  right  hand ;  and  in  reading  them,  to  begin  at 
the  left  hand. 

2.  Since  the  figures  in  the  first  or  right  hand  period  always  denote 
units,  the  name  of  the  period  is  not  pronounced.  Hence,  in  reading 
figures,  when  no  period  is  mentioned,  it  is  always  understood  to  be 
the  right  hand,  or  units'  period. 

EXERCISES   IN   NUMERATION. 

Note. — At  first  the  pupil  should  be  required  to  apply  to  each  figure 
the  name  of  the  place  which  it  occupies.  Thus,  beginning  at  the 
right  hand,  he  should  say,  "  Units,  tens,  hundreds,"  <xc.,  and  point 
at  the  same  time  to  the  figure  standing  in  the  place  which  he  men- 
tions. It  will  be  a  profitable  exercise  for  young  scholars  to  write  the 
examples  upon  their  slates  or  paper,  then  point  them  off  into  periods, 
and  read  them. 


QUEST. — 13.  How  do  you  read  numbers  expressed  by  figures? 
Obs.  Where  begin  to  point  them  off?  Where  to  read  them  ?  Do  you 
pronounce  the  name  of  the  right  hand  period  ?  When  no  period  is 
named,  what  is  understood  ?  14.  In  the  French  method  of  numera- 
tion, how  many  figures  are  there  in  a  period  ?  How  many  in  the 
English  method  ?  Which  method  is  preferable  ? 


ARTS.  13,  14.] 


NUMERATION. 


Read  the  following  numbers : 


Ex,  1. 

127 

11.     75407 

21.     5604700 

2. 

172 

12.    125242 

22.     2020105 

3. 

721 

13.    240251 

23.    45001003 

4. 

520 

14.    407203 

24.    30407045 

5. 

603 

15.    300200 

25.   145560800 

6. 

4506 

16.   1255673 

26.     8900401 

7. 

7045 

17.   5704086 

27.   250708590 

8. 

8700 

18.    207047 

28.   803068003 

9. 

25008 

19.   2605401 

29.  2175240670 

10. 

40625 

20.   4040680 

30.  7240305060 

31. 

45290100300 

36.    13657240129698 

32. 

160000050000 

37.    98609006006906 

33. 

7005003007 

38.    80079401697000 

34. 

101279200361 

39.   167540000000465 

35. 

1143206000675 

40.   504069470300400 

14.  The  method  of  dividing  numbers  into  periods  of 
three  figures,  is  the  French  Numeration.  The  English 
divide  numbers  into  periods  of  six  figures.  The  French 
method  is  the  more  simple  and  convenient.  It  is  gene- 
rally used  throughout  the  continent  of  Europe,  as  well  as 
in  America,  and  has  been  recently  adopted  by  some  Eng- 
lish authors. 

EXERCISES   IN   NOTATION. 

Write  the  following  numbers  in  figures  : 

1.  Twenty-seven.     Ans.  27. 

2.  Seventy-two.     Ans.  72. 

3.  One  hundred  and  twenty-five. 

4.  Three  hundred  and  fifty-two. 

5.  Two  hundred  and  four.     Ans.  204. 

6.  One  thousand  and  forty-two.     Ans.   1042. 

7.  Thirty  thousand  nine  hundred  and  seven. 

Ans.  30907. 

OBS.  It  will  be  observed,  that  in  the  5th  example  no  tens  are  men- 
tioned, in  the  6th  no  hundreds,  and  that  these  places  in  the  answers 
ar*»  filled  by  ciphers.  In  all  cases,  when  any  intervening  order  i» 
onaitted  in  the  given  example,  the  place  of  that  order  in  the  answer 
tnust  be  filled  by  a  cipher.  Hence, 


22  NUMERATION.  [SECT.  1. 

1  5  •  To  express  numbers  by  figures. 


at  the  left  hand,  and  write  in  each  ordei  the  figure 
cohich  denotes  the  given  number  in  that  order. 

If  any  intervening  orders  are  omitted  in  the  proposed  num- 
ber, write  ciphers  in  their  places. 

8.  Forty-six  thousand  and  four  hundred. 

9.  Ninety-two  thousand,  one  hundred  and  eight. 

10.  Sixty-eight  thousand  and  seventy. 

11.  One  hundred  and  twenty-four  thousand,  six  hun- 
dred and  thirty. 

12.  Two  hundred  thousand,  one  hundred  and  sixty. 

13.  Four  hundred  and  five  thousand,  and  forty-five. 

14.  Three  hundred  and  forty  thousand. 

15.  Nine  hundred  thousand,  seven  hundred  and  twenty. 

16.  One  million,  and  seven  hundred  thousand. 

1  7.  Thirty  -six  millions,  twenty  thousand,  one  hundred 
and  fifty. 

18.  One  hundred  millions,  and  forty-five. 

19.  Mercury  is  thirty-seven  millions  of  miles  from  the 
sun. 

20.  Venus,  sixty-nine  millions. 

21.  The  Earth,  ninety-five  millions. 

22.  Mars,  one  hundred  and  forty-five  millions. 

23.  Jupiter,  four  hundred  and  ninety-four  millions. 

24.  Saturn,  nine  hundred  and  seven  millions. 

25.  Herschel,  one  billion,  eight  hundred  and  ten  mill- 
ions. 

26.  Seven  billions,  nine  hundred  millions,  and  forty 
thousand. 

27.  Sixty  billions,  seven  millions,  and  four  hundred. 

28.  One  hundred  and  thirteen  billions,  six  hundred  and 
fifty  thousand. 

29.  Four  hundred  and  six  billions,  eighty  millions,  and 
seven  hundred. 

30.  Twenty-five  trillions,  and  ten  thousand. 


QUEST. — 15.  How  are  numbers  expressed  by  figures  ?     If  any  inter 
veiling  order  is  omitted  in  the  example,  how  is  its  place  supplied  ? 


AETS.  13,  16.]  ADDITION.  33 

SECTION    II. 
ADDITION. 

MENTAL     EXERCISES. 

ART.  J  O  Ex.  1.  George  bought  a  slate  for  9  cents, 
a  sponge  for  6  cents,  and  a  pencil  for  1  cent :  how  many 
cents  did  he  pay  for  all  ? 

OBS.  To  sol''*  this  example,  we  must  add  together  the  number  of 
cents  which  he  paid  for  the  several  articles.  Thus,  9  cents  and  6 
cents  are  15  cents,  and  one  cent  more  makes  16  cents.  Ans.  He  paid 
i6  cents. 

2.  Henry  gave  8  cents  for  a  writing-book,  6  cents  for 
an  inkstand,  and  4  cents  for  some  quills  :  how  many  cents 
did  he  give  for  all  ? 

3.  Sarah  obtained  4  credit  marks  yesterday,  3  the  day 
before,  and  5  to-day  :  how  many  credit  marks  has  she 
in  all  ? 

4.  John  had  6  peaches,  and  his  mother  gave  him  10 
more  :  how  many  peaches  had  he  then  ? 

5.  Harriet  has  7  pins  ;  she  has  given  away  4,  and  lost 
2  :  how  many  pins  had  she  at  first  ? 

6.  If  a  quart  of-  cherries  is  worth  5  cents,  a  pound  of 
figs  9  cents,  and  a  lemon  4  cents,    how  much  are  they  all 
worth  ? 

7.  Joseph  paid  6  cents  for  some  raisins,  7  cents  for  a 
top,  and  3  cents  for  some  fish-hooks  :  how  many  cents  did 
he  pay  for  all  1 

8.  Mary  has  9  white  roses  and  8  red  ones  :  how  many 
loses  has  she  in  all  ? 

9.  A  beggar  met  four  men,  one  of  whom  gave  him  3 
shillings,  another   2,  another  1,  and  the  last  5  shillings: 
how  many  shillings  did  the  beggar  receive  ? 

10.  A  farmer  sold  4  bushels  of  apples  to  one  customer, 
R  to  another,  5  to  a  third,  and  2  to  a  fourth  :  how  many 
oushels  did  he  sell  to  all  1 


£4 


ADDITION. 


[SECT.  IL 


ADDITION  TABLE. 


2  and 

3  and 

4  and 

5  and 

6  and 

7  and 

8  and    i   9  and 

lare   3 

1  are    4 

lare   5 

1  are    6 

lare    7 

1  are    8 

1  are   9'  1  are  10 

2    "     4 

2    "     5 

2 

6 

2    "     7 

2 

8 

2 

9 

2 

JO 

2 

11 

3    »     5 

3    "     6 

3 

7 

3 

8 

3 

9 

3 

10 

3 

11 

3 

12 

4    "     6 

4    «     7 

4 

8 

4 

9 

4 

10 

4 

11 

4 

12 

4 

13 

5 

7 

5    "     8 

5 

9 

5 

10 

5 

11 

5 

12 

5 

13 

5 

14 

6 

8 

6    "     9 

6 

10 

6 

11 

6 

12 

6 

13 

6 

14 

6 

15 

7 

9 

7    "10 

7 

11 

7 

12 

7    "   13 

7 

14 

7 

15 

7 

16 

8 

10 

8    "   11 

8 

12 

8 

13 

8    "   14 

8 

15 

8 

16 

8 

17 

9 

11 

9    "   12 

9 

13 

9 

14 

9    "   15 

9 

16 

9 

17 

9 

18 

10 

12 

10    "    13 

10 

14 

10 

15 

10    "    16 

10 

17 

10 

18 

10 

19 

Note. — It  is  an  interesting  and  profitable  exercise  for  young  pupils 
to  recite  tables  in  concert.  But  it  will  not  do  to  depend  upon  this 
method  alone.  It  is  indispensable  for  every  scholar  who  desires  to  be 
accurate  either  in  arithmetic  or  business,  to  have  the  common  arith- 
metical tables  distinctly  and  indelibly  fixed  in  his  mind.  Hence,  af 
ter  a  table  has  been  repeated  by  the  class  in  concert,  or  individually, 
the  teacher  should  ask  many  promiscuous  questions,  to  prevent  its 
being  recited  mechanically,  from  a  knowledge  of  the  regular  increase 
of  numbers. 

Ex.  11.  How  many  are  12  and  10?  22  and  10?  32 
and  10?  42  and  10?  52  and  10?  62  and  10?  72  and  10? 
82  and  10?  92  and  10? 

12.  How  many  are  24  and  10?  36  and  10?  48  and 
10?  53  and  10?  67  and  10?  91  and  10?  86  and  10? 

78  and  10?  69  and  10?  97  and  10? 

13.  How  many  are  19  and  4  ?  29  and  4  ?  39  and  4  ? 

79  and  4?  59  and  4?  89  and  4?  99  and  4?  69  and  4? 
49  and  4  ? 

14.  How  many  are  17  and  8  ?  27  and  8  ?  47  and  8  1 
67  and  8  ?  57  and  8  ?  97  and  8  ?  87  and  8  ? 

15.  How  many  are  16  and  7  ?  26  and  7  ?  56  and  7  ? 
36  and  7  ?  76  and  7  ?  96  and  7  ? 

16.  How  many  are  14  and  6?  24  and  6?  84  and  63 
74  and  6  ?  54  and  6  ?  64  and  6  ?  94  and  6  ? 

1 7.  Add  2  to  itself  till  the  sum  is  a  hundred. 

OBS.  This  and  the  next  four  examples  may  be  recited  in  concert 
Thus,  2  and  2  are  four,  and  2  are  6,  and  2  are  8,  &c. 

18.  Add  3  in  the  same  manner,  till  the  sum  is  a  hun 
dred  and  two. 

19.  Add  5  in  the  same  manner,  till  the  sum  is  a  hiu/ 
dred  and  tea 


ART.  16.]  ADDITION.  25 

20.  Add  4  in  the  same  manner,  till  the  sum  is  a  hun- 
dred and  twelve. 

21.  Add  10  in  the  same  manner,  till  the  sum  is  a  hun- 
dred and  twenty. 

22.  A  man  bought  a  sheep  for  3  dollars,  a  cow  for  21 
dollars,  and  a  calf  for  5  dollars :  how  much  did  he  pay 
for  the  whole. 

23.  A  shopkeeper  sold  a  dress  to  a  lady  for  15  dollars, 
a  muff  for  10  dollars,  and  a  bonnet  for  6  dollars :  what 
was  the  amount  of  her  bill  ? 

24.  A  drover  bought  16  sheep  of  one  farmer,  9  of  an- 
other, 10  of  another,  and  6  of  another  :  how  many  sheep 
4id  he  buy  ? 

25.  Harry  gave  31  cents  for  his  arithmetic,  10  cents 
for  a  writing-book,  8  cents  for  a  ruler,  and  6  cents  for  a 
lead  pencil :  how  many  cents  did  he  pay  for  all  ? 

26.  What  is  the  sum  of  10  and  12  and  5  and  4  ? 

27.  William  bought  a  pair  of  boots  for  26  shillings, 
•md  a  cap  for  9  shillings :  how  many  shillings  did  he  give 
Tor  both  ? 

28.  Susan  bought  a  comb  for  17  cents,  a  purse  for  8 
cents,  and  a  spool  of  cotton  for  5  cents :  how  much  did 
she  pay  for  all  ? 

29.  A  farmer  sold  a  ton  of  hay  for  18  dollars,  a  cow 
for   10  dollars,  and  a  cord  of  wood  for  3  dollars :  how 
much  did  he  receive  for  all  ? 

30.  A  merchant  sold  15  barrels  of  flour  to  one  man,  5 
to  another,  and  7  to  another  :  how  many  barrels  of  flour 
did  he  sell  ? 

31.  In  a  certain  school  there  are  60  boys,  and  30  girls: 
how  many  scholars  does  that  school  contain  ?. 

Analysis.— 60  is  6  tens,  and  30  is  3  tens;  (Art.  7. 
Obs.  2  ;)  6  tens  and  3  tens  are  9  tens,  and  9  tens  are  90. 

Ans.  90  scholars. 

32.  A  mechanic  sold  a  wagon  for  30,  and  a  sleigh  fo* 
20  dollars  :  how  much  did  he  get  for  both  ? 

33.  40  is  how  many  tens?  60?  20?  30?  70?  80? 
50?  90?  100? 


26  ADDITION.  [SECT.  II. 

34.  6  tens  are  how  many  ?  8  tens?  9  tens?  10  tens? 
11  tens?   12  tens?   13  tens?   14  tens?   15  tens?   16  tens? 
17  tens?   18  tens  ?   19  tens  ?  20  tens? 

35.  7  tens  and  2  tens  are  how  many?  Ans.  9  tens, 
or  90. 

36.  8  tens  and  3  tens  are  how  many  ?  5  tens  and  8 
tens  ?  7  tens  and  8  tens  ?  6  tens  and  9  tens  ?  9  tens  and 

8  tens?  10  tens  and  6  tens? 

37.  In  a  certain  orchard  there  are  80  apple-trees,  and 
40  peach-trees :  how  many  trees  does  the  orchard  con- 
tain? 

38.  A  traveler  rode  90  miles  in  the  cars,  and  60  miles 
in  stages  :  how  many  miles  did  he  travel  ? 

39.  A  man  gave  60  dollars  for  his  horse,  30  dollars  for 
his  harness,  and  20  dollars  for  his  cart :  how  much  did 
he  pay  for  all  ? 

40.  A  man  bought  a  horse  for  98  dollars,  and  a  wagon 
for  65  dollars  :  how  much  did  he  give  for  both  ? 

Analysis. — 98  is  composed  of  9  tens  and  8  units,  and 
65  is  composed  of  6  tens  and  5  units.     (Art.  7.  Obs.  3.) 

9  tens  and  6  tens  are  15  tens,  or  1  hundred  and  5  tens  ; 
8  units  and  5  units  are   13  units,  or  1  ten  and  3  units ; 
now  1  ten  added  to  5  tens,  makes  6  tens  or  60,  and  3 
units  are  63,  which,  joined  with  the  hundred,  makes  163. 

Ans.  He  paid  163  dollars. 

41.  How  many  are  63  and  24?  Ans.  87. 

42.  How  many  are  68  and  25  ? 

43.  How  many  are  56  and  23  and  5  ? 

44.  How  many  are  83  and  72  and  4  and  6  ? 

45.  How  many  are  72  and  25  and  10  and  2? 

46.  Bought  a  pound  of  tea  for  60  cents,  an  ounce  o! 
pepper  for  8  cents,  and  a  quart  of  molasses  for  10  cents 
what  does  my  bill  amount  to  ? 

47.  The  price  of  a  geography  is  55  cents,  and  the  price 
of  a  grammar  is  42  cents :  what  is  the  cost  of  both  ? 

48.  Paid  7  dollars  for  a  barrel  of  flour,  17  dollars  for  a 
ton  of  hav,  and  30  dollars  for  a  cow :  what  is  the  cost  of 
all? 

49.  In  January  there  are  31  days,  and  in  February  28 
days :  how  many  days  are  there  in  both  months  ? 


ARTS.  17-20.]  ADDITION.  27 

50.  A  man,  having  three  sons,  gave  50  dollars  to  the 
oldest,  40  dollars  to  the  second,  and  30  dollars  to  the 
youngest :  how  many  dollars  did  he  give  to  the  three  ? 

17.  The  learner  Avill  perceive  that  the  solution  of  each 
of  the  preceding  examples,  consists  in  finding  a  single 
number  which  will  exactly  express  the  value  of  the  several 
given  numbers  united  together. 

18.  The  process  of  wiling  two  or  more  numbers  together, 
so  as  to  form  one  single  number,  is  called  ADDITION. 

The  answer,  or  the  number  thus  found,  is  called  the 
mm  or  amount. 

OBS.  When  the  numbers  to  be  added  are  all  of  the  same  denomi- 
nation, as  all  dollars,  or  all  pounds,  &c.,  the  operation  is  called  Simple 
Addition. 

1 9.  Signs. — Addition  is  often  represented  by  the  sign 
(+),  which  is  called  plus.     It  consists  of  two  lines,  one 
horizontal,  the  other  perpendicular,  forming  a  cross,  and 
shows  that  the  numbers  between  which  it  is  placed,  are 
to  be  added  together.     Thus  the  expression  6+8,  signi- 
fies that  6  is  to  be  added  to  8.     It  is  read,  "  6  plus  8," 
or  "  6  added  to  8." 

Note.— Plus  is  a  Latin  word,  originally  signifying  "more,"  hen 36 
"  added  to." 

20.  The  equality  between  two  numbers,  or  sets  of 
numbers,  is  expressed  by  two  parallel  lines  (=),  called 
the  sign  of  equality.     It  shows  that  the  numbers  between 
which  it  is  placed  are  equal  to  each  other.     Thus  the  ex- 
pression 5-f-3=8,  denotes  that  5  added  to  3  are  equal  to  8. 
It  is  read,  "  5  plus  3  equal  8,"  or  "  the  sum  of  5  plus  3 
is  equal  to  8."     So  7+5=8+4=12. 


Q.— 18.  What  is  addition  ?  What  is  the  answer  called  ?  Obs.  When 
the  numbers  to  be  added  are  all  of  the  same  denomination,  what  is  the 
operation  called  ?  19.  What  is  the  sign  of  addition  called  I  Of  what 
does  it  consist  ?  What  does  it  show  ?  Note.  What  is  the  meaning  ol 
the  word  plus  ?  20.  How  is  the  equality  between  two  numbers  repre- 
sented ?  What  does  the  sign  of  equality  show  ?  How  is  the  expres 
sion  54-3=8.  read  ?  How,  7-f-5=*8-H=l2 ? 


ADDITION.  [SECT.  II 


EXERCISES   FOR   THE    SLATE. 

2 1  •  Examples  in  which  the  numbers  to  be  added  are 
small,  should  be  solved  mentally  ;  but  when  the  numbers 
are  large,  the  operation  may  be  facilitated  by  setting  them 
down  upon  a  slate,  or  black-board.  The  manner  of  doing 
this  will  now  be  explained. 

OBS.  Pupils  not  ^infrequently  seem  to  infer,  that  when  they  take 
up  the  slate  and  pencil,  they  can  lay  aside  thinking ;  that  the  hands 
are  to  solve  the  question  without  the  aid  of  the  intellect.  Hence 
operations  upon  the  slate  are  often  a  merely  mechanical  effort,  as 
listless  and  mindless  as  the  talking  of  a  parrot,  or  the  trudging  of  a 
dray-horse.  This  is  a  sad  mistake.  It  is  sure  to  render  the  study  of 
arithmetic  irksome,  and  to  destroy  the  progress  of  the  learner. 

It  is  not  the  object  in  using  the  slate  to  supersede  thinldng  and  rear 
soning,  but  to  assist  the  memory  in  retaining  the  numbers  and  the 
several  steps  of  the  operation,  while  the  intellect  is  carrying  on  the 
process  of  thinking  and  reasoning. 

The  hands  simply  write  down  the  figures  or  the  result  of  the  ope- 
ration, but  it  is  the  mind,  and  the  mind  only,  that  performs  the  addi- 
tion and  all  other  arithmetical  calculations,  whether  we  use  the  slate 
or  not.  Hence,  whoever  wishes  to  become  a  proficient  in  arithmetic, 
must  never  allow  his  mind  to  become  inactive  when  using  his  slate, 
nor  pass  a  single  solution  without  understanding  the  reason  of  the 
several  steps. 

Ex.  1.  A  man  bought  a  pound  of  tea  for  63  cents,  and. 
a  pound  of  coffee  for  24  cents  :  how  much  did  he  pay  foi 
both? 

Directions. — Write  the  numbers  Operation. 

under   each    other,  so   that   units 
may  stand  under  units,  tens  under      a  'g 
tens,  and  draw  a  line  beneath  them.      ®  & 
Then,  beginning  at  the  right  hand      6  3  price  of  tea. 
or  units,  add  each   column  scpa-     24      "      of  coffee. 

rately  in  the  following  manner :  4     

units  and  3  units  are  7  units.—      8  7  cts.  price  of  both, 


QUEST. — 21.  How  should  examples,  in  which  the  numbers  to  be  add- 
ed are  small,  be  solved  ?  When  they  are  large,  how  may  the  operation 
be  facilitated  ?  Obs.  Is  the  slate  designed  to  supersede  thinking  and 
reasoning  ?  What  is  its  use  ?  How  are  all  arithmetical  calculations 
performed  ?  What  direction  is  given  to  those  who  wish  to  become 
proficient*  in  arithmetic  ? 


ARTS.  21, 22.]  ADDITION.  29 

Write  the  7  in  units'  place,  under  the  column  added. 

2  tens  and  6  tens  are  8  tens.     Write  the  8  in  tens'  place. 
The  amount  is  87  cents. 

jy0te. — The  learner  will  perceive  that  the  operation  upon  the  slate 
is  essentially  the  same  as  the  mental  solution  of  the  same  question ; 
(Art.  16.  Ex.  41  ;)  and  that  both  give  the  same  result. 

2.  A  butcher  purchased  two  droves  of  sheep,  the  first 
containing  436,  and  the  second  243 :  how  many  sheep 
did  both  droves  contain  ? 

Write  the  numbers  under  each          O    •    t' 
other,  and  proceed  as  before.    Thus,  ™ 

3  units  and  6  units  are  9  units ;  4  436  First  drove, 
tens  and  3  tens  are  7  tens ;  2  hun-  243  Second  " 
dreds  and  4  hundreds  are  6  hun-  

dreds.     The  amount  is  679.  679  Ans. 

22.  It  will  be  perceived,  from  these  examples,  that 
units  are  added  to  units,  tens  to  tens,  and  hundreds  to  hun* 
dreds;  that  is,  figures  of  the  same  order  are  added  to  each 
other.  This  is  the  only  way  numbers  can  be  added. 
For,  figures  standing  in  different  orders  or  columns,  ex- 
press different  values ;  (Art.  8 ;)  consequently,  if  united 
together  in  a  single  sum,  the  amount  can  neither  be  of 
one  order  nor  another.  Thus,  3  units  and  3  tens  will 
neither  make  six  units,  nor  six  tens,  any  more  than  3  or- 
anges and  3  apples  will  make  6  apples,  or  6  oranges.  In 
like  manner  it  is  plain  that  4  tens  and  4  hundreds  will 
neither  make  8  tens,  nor  8  hundreds. 

OBS.  In  writing  numbers  to  be  added,  great  care  should  be  taken 
to  place  units  under  units,  tens  under  tens,  &c.,  in  order  to  prevent 
mistakes  which  would  otherwise  be  liable  to  occur  from  adding  differ- 
ent orders  to  each  other. 

3.  A  man  found  two  purses  of  money,  one  containing 
425  dollars,  the  other  361  dollars:  how  many  dollars  did 
both  purses  contain  ? 

QUEST. — In  the  1st  example  how  do  you  write  the  numbers  for  addi- 
tion ?  Which  column  do  you  add  first?  Which  next?  Note.  Does  the 
operation  upon  the  slate  differ  from  the  mental  solution  of  the  same 
question  1  22.  Can  figures  standing  in  different  orders  be  added  to 
each  other  ?  Why  not  ?  Illustrate  by  an  example.  Oba.  What  is  th« 
object  in  writing  unite  under  units,  &<s  ? 


30  ADDITION.  [SECT.  II, 

4.  What  is  the  sum  of  3261  and  5428? 

5.  What  is  the  sum  of  45436  and  12321? 

6.  What  is  the  sum  of  420261  and  231204? 

7.  What  is  the  sum  of  3021040  and  5630721  ? 

8.  What  is  the  sum  of  730043000  and  268900483? 
Write  the  following  examples  upon  the  slate,  and  find 

the  sum  of  each: 

9.  10.                 11.  12. 

221  4212  62022  82202310 

345  3120  5103  3060231 

422  341  21640  617403 


23.  When  the  sum  of  a  column  does  not  exceed  9,  it 
must  be  written,  as  we  have  seen,  under  the  column  add- 
ed. But  when  the  sum  of  a  column  exceeds  9,  it  requires 
two  or  more  figures  to  express  it ;  (Art.  7  ;)  consequently, 
it  cannot  all  be  written  under  the  column  added.  What 
then  must  be  done  ?  We  will  now  illustrate  this  case. 

13.  A  man  paid  98  dollars  for  a  horse,  and  65  dollars 
for  a  wagon:  how  much  did  he  pay  for  both? 

Directions. — Write  the  numbers,  ^ 

and  begin  at  the  right  hand,  as  be-      Qs  Uper  'M0f\ 
fore.     Thus  5  units  and  8  units  are      ^  Pf  °ff  h°rse* 
13  units.     Now  13  is  1  ten  and  3    __^  of  wagon, 

units,  and  requires  two  figures  to     ,,,„ 
express  it;  (Art.  7;)  consequently    163  Amount 
it  cannot  be  written  under  the  column  of  units.     Hence 
we  write  the  3  units  in  the  units'  place,  and  reserving  the 
1  ten  or  left  hand  figure  in  the  mind,  add  it  with  the  tens 
in  the  next  column.     Thus  1  ten  (which  was  reserved) 
and  6  tens  are  7  tens,  and  9  are  16  tens,  which  are  equal 
to  1  hundred  and  6  tens.     Write  the  6  tens  under  the  col* 
umn  added,  and  the  1  hundred  in  the  place  of  hundreds. 
The  amount  is  163  dollars. 

QUEST. — 23.  When  the  sum  of  a  column  does  not  exceed  9,  where  is 
it  written  ?  Can  the  whole  sum  be  written  under  the  column  when  it 
exceeds  9  ?  Why  not  ?  In  the  13th  example,  what  is  the  sum  of  the 
units'  column  ?  How  do  you  dispose  of  it  ?  What  do  you  do  with  th# 
Burn  of  the  next  eolumn  ? 


23-25.]  ADDITION.  31 

OBS.  It  will  be  perceived  that  the  operation  upon  the  slate  is  sub- 
stantially the  same  as  the  mental  solution  ot*  ths  same  question.  (Art 
16.  Ex.  40.)  In  each  case,  we  add  the  orders  separately ;  in  each, 
iinding  the  sum  of  the  unit's  column  to  be  13,  or  1  ten  and  3  units, 
we  add  the  1  ten  to  the  number  of  tens  which  is  contained  in  the 
example ;  and  in  each  we  obtain  the  same  result. 

1 4.  A  gentleman  bought  a  span  of  horses  for  645  dol- 
ars,  a  carriage  for  467  dollars,  and  a  set  of  harness  for 
158  dollars  :  how  much  did  he  give  for  the  whole  estab- 
lishment ? 

Proceed    as    before.      Thus    8  Operation. 

units  and  7  units  are  15  units,  or  645  price  of  horses, 
we  simply  say,  8  and  7  are  15.  467  "  carriage, 
and  5  are  20.  Set  the  0  under  158  "  harness. 

the    column    added,    and,    reserv-     

ing  the  2,  add  it  with  the  next  1 270  dollars.  Ans. 
column..  2  (which  was  reserved)  and  5  are  7,  and  6  are 
13.  and  4  are  17.  Set  the  7  under  the  column  added, 
and  add  the  1  with  the  next  column.  1  (which  was 
reserved)  and  1  are  2,  and  4  are  6,  and  6  are  12.  Set  the 
2  under  the  column  added,  and  since  there  is  no  other 
column  to  be  added,  write  the  1  in  the  next  place  on  the 
left.  The  amount  is  1270  dollars. 

24.  The  process  of  reserving  the  tens  or  left  hand 
figure,  when  the  sum  of  a  column  exceeds  9,  and  adding 
it  mentally  to  the  next  column,  is  called  carrying  tens. 

25.  When  the  sum  of  the  column  exceeds  9,  set  the 
units  or  right  hand  figure  under  the  column  added,  and 
carry  the  tens  or  left  hand  figure  to  ^he  next  column.     In 
adding  the  last  column  on  the  lelt,  it  will  be  noticed  we 
set  down  the  whole  sum.     This  is  done  for  the  obvious 
reason  that  there  are  no  figures  in  the  next  column  to 
which  the  left  hand  figure  can  be  added,  and  is  in  fact 
carrying  it  to  the  next  order. 


QUEST. — Obs.  Does  the  operation  upon  the  slate  differ  essentially 
from  the  mental  solution  of  the  same  example  ?  In  what  respects  do 
they  coincide  ?  24.  What  is  the  process  of  reserving  the  tens  and  ad- 
ding them  to  the  next  column,  called  *  '-£5.  When  the  sum  of  any  col- 
umn exceeds  9,  what  is  to  be  done  with  it  ?  When  the  sura  is  20, 
*hat  do  you  set  down,  and  what  do  you  carry  ?  If  27,  what  ? 


S3  ADDITION.  [SECT.  II. 


ILLUSTRATION  OF  THE  PRINCIPLE  OF  CARRYING. 

26.  To  illustrate  the  principle  of  carrying,  let  us  take 
the  thirteenth  example,  and  as  we  add  the  columns,  write 
down  the  whole  sum  of  each  in  a  *  ,. 

separate    line.       The   sum   of  the      no  OPemtwf^ 
units'  column  is  13  units,  or  1  ten      ^  P™e  f  horse' 
and  3  units  ;  the  sum  of  the  tens'  wa£on' 

column  is   15  tens,  or   1  hundred 


and  5   tens.      Now   adding    these 

results  together  as  they  stand,  i.  e.  ens' 

adding  units  to  units,  tens  to  tens,    771 

&c,  the  amount  is  163,  the  same     163  Amount' 

as  before.     Thus,  it  will  be  seen  that  the  1  ten  or  left 

hand  figure  in  the  sum  of  the  first  column,  is  added  to  the 

sum  of  the  next  column  or  the  15  tens,  in  the  same  man 

ner  as  it  was  in  the  solution  above. 

Again,  the  principle  of  carrying  may  be  illustrated  by 
separating  the  numbers  to  be  added  into  the  parts  or  or 
debt's  of  which  they  are  composed.  Thus, 

98  is  composed  of  9  tens  or  90,  and  8  units. 
65  6  tens  or  60,  and  5  units. 

150  and  13. 

Adding  the  sum  of  the  units  (13)       13 
to  the  sum  of  the  tens,  (150)       15& 

the  amount  is     163 
Take  also  the  fourteenth  example  : 

645  is  composed  of  600,      40  and  5  units. 

467  "  400,      60  and  7  units. 

'  158  100,      50  and  8  units. 

1270  Amount.  1100    150  and  20 


QuEar  —If  the  sura  is  36,  what  ?  If  70,  what  ?  What  do  you  do  wit; 
the  sum  of  the  left  hand  column  ?  Why  1  Does  this  differ  from  carrying 


ARTS.  SO,  27.]  ADDITION. 

.  ,j.       ,,  IA         .,   }  1100  sum  of  hun 

Adding  these  results,  units  f    150    «    of  tens 

to  units,  tens  to  tens,  &c,          t      2Q     «    Qf 


we  have   1270  Amount. 

Here  it  will  also  be  noticed,  that  when  the  sum  of  any 
column  exceeds  9,  the  tens  or  left  hand  figure  is  added,  in 
every  instance,  to  the  same  column  or  order  to  which  it  is 
carried  in  the  solution. 

27.  From  these  illustrations  it  will  t>e  seen,  that  the 
process  of  carrying  tens  is,  in  effect,  simply  adding  the  tens 
to  tens,  the  hundreds  to  hundreds,  &c.,  which  are  con- 
tained in  the  given  example ;  or  adding  figures  of  the 
same  order  together,  which  is  the  only  way  they  can  be 
added.  (Art.  22.)  For,  if  the  sum  of  any  column  ex- 
ceeds 9,  and  thus  requires  two  or  more  figures  to  express 
it,  (Art.  7,)  the  right  hand  figure  denotes  units  of  the 
same  order  as  the  column  added,  arid  the  left  hand  figure 
denotes  units  of  the  next  higher  order  ;  (Art.  8  ;)  conse- 
quently, it  is  of  the  same  order  as  the  next  column  to 
which  it  is  carried.  The  result  will  obviously  be  the 
same,  whether  we  add  the  tens  in  their  proper  place,  as 
we  proceed  in  the  operation,  or  reserve  them  till  we  have 
added  the  respective  columns,  and  then  add  them  to  the 
same  orders.  The  former  method  is  the  more  convenient 
and  expeditious,  and  is  therefore  adopted  in  practice. 

15.  What  is  the  sum  of  473  and  987?  Ans.  1460. 

16.  17.  18.  19. 

4674  67375  84056  405673 

6206  87649  5721  720021 

4321  6048  41630  369115 

8569  452  163  505181 


QUEST.— 27.  What,  in  effect,  is  the  process  of  carrying  the  tens  to 
the  next  column  ?  How  does  tliis  appear  \  Does  it  make  any  difference 
with  the  result,  when  the  tens  are  added  to  the  next  column!  When 
are  they  commonly  added  ?  Why  ? 


ADDITION.  [SECT.  II. 

28.  PROOF. — Beginning  at  the  top,  add  each  column 
downwards,  and  if  the  second  result  is  the  same  as  the  first, 
ike  work  is  supposed  to  be  right. 

OBS.  The  object  of  beginning  at  the  top  and  adding  downwards,  la 
that  the  figures  may  be  taken  in  a  different  order  from  that  in  which 
they  were  added  before  j  otherwise,  if  a  mistake  has  been  made  the 
first  time  adding,  we  should  be  liable  to  fall  into  the  same  again.  But 
the  order  being  reversed,  the  presumption  is,  that  any  mistake  which 
may  have  been  made  will  thus  be  detected ;  for  it  can  hardly  be  sup- 
posed that  two  mistakes  exactly  equal  will  occur. 

20.  Find  the  sum  of  256,  763,  and  894,  and  prove  the 
operation. 

21.  Find  the  sum  of  8054.  5730,  and  3056,  and  prove 
the  operation. 

22.  Find  the  sum  of  74502,  83000,  and  62581,   and 
prove  the  operation. 

23.  Find  the  sum  of  68056,  31067,  680,  and  200,  and 
prove  the  operation. 

24.  Find  the  sum  of  50563,  8276,  75009,31,  and  856, 
and  prove  the  operation. 

25.  Find  the  sum  of  65031,2900,  35221,  and  870,  and 
prove  the  operation. 

29.  From  the  preceding  illustrations  and  principles 
we  derive  the  following 

GENERAL  RULE  FOR  ADDITION. 

I.  Write  the  numbers  to  be  added  under  each  other,  so 
that  units  may  stand  under   units,   tens  under  tens^  fyc. 
(Art.  21,  Ex.  1.) 

II.  Begin  at  the  right  hand,  and  add  each  column  sepa- 
rately.    When  the  sum  of  a  column  does  not  exceed  9,  write 
it  under  the  column ;  but  if  the  sum  of  a  column  exceeds  9. 
write  the  units'  figure  under  the  column  added,  and  carry  tht 
lens  to  the  next  column.  (Arts.  23,  25.) 


QUEST. — 28.  How  is  addition  proved  ?  Obs.  Why  add  the  column* 
downwards,  instead  of  upwards  ?  29.  What  is  the  general  nils  roi  «** 
eition? 


AETS.  28,  29.]  ADDITION.  95 

III.  Proceed  in  this  manner  through  all  the  orders,  ana 
finally  set  down  the,  whole  sum  of  the  last  or  the  left  hand 
column.  (Art.  25.) 

EXAMPLES   FOR   PRACTICE. 

1.  A  man  bought  a  quantity  of  flour  for  38  dollars,  a 
ton  of  hay  for  14  dollars,  and  a  firkin  of  butter  for  12 
doll'-.i.'s.     How  much  did  he  give  for  the  whole? 

2.  A  grocer  bought  three  boxes  of  honey ;    the  first 
contained   22   pounds,  the  second    15,  and   the  third  9 
pounds.     How  many  pounds  were  there  in  all? 

3.  A  man  being  asked  his  age,  answered  that  it  was 
equal  to  the  united  ages  of  his  three  children,  the  oldest 
of  whom  was  18,  the  second  16,  and  the  third  14  years 
old.     What  was  his  age  ? 

4.  A  man  bought  5  hogsheads  of  molasses  for  238  dol- 
lars, and  sold  it  so  as  to  gain  75  dollars.     How  much  did 
he  sell  it  for  ? 

5.  A  lady  purchased  -  materials  for  3  dresses ;   for  the 
first  she  paid  15  dollars,  for  the  second,  9  dollars,  and  for 
the  third,  7  dollars.     How  much  did  she  pay  for  them  ail  ? 

6.  A  boy  bought  a  cap  for  12  shillings,  a  pair  of  gloves 
for  6  shillings,  a  pair  of  boots  for  16  shillings,  and  a  book 
for  6  shillings.     How  much  did  he  give  for  the  whole  ? 

7.  A  gentleman  owns  3  houses ;  for  the  first   he  re- 
ceives a  rent  of  150  dollars,  for  the  second  175,  and  for  the 
third  225  dollars.     What  is  the  sum  of  all  his  rents  ? 

8.  A  shopkeeper  commenced  business  with  1530  dol- 
lars ;  after  trading  some  time,  he  found  he  had  gained  950 
dollars.     How  much  >ad  he  then  ? 

9.  A  man  bought  a  horse  for  87  dollars,  a  carriage  for 
75  dollars,  and  a  harness  for  28  dollars.     How  much  did 
he  give  for  the  whole  ? 

10.  What  number  of  dollars  are  there  in  four  purses; 
the  first  containing  25  dollars,  the  second  73,  the  third  84 
and  the  fourth  96  dollars  ? 

1 1.  A  poor  man  having  lost  his  house  by  fire,  to  help 
him  repair  his  loss,  one  man  gave  him  25  dollars,  another 
15,  anofher  10,  another  5,  and  another  3.     How  much 
^td  ho  reativt  from  all? 


86  ADDITION.  [SECT.  I 

12.  In  a  certain  school  there  were  three  classes  in 
arithmetic ;  the  first  class  contained  8  scholars,  the  second 
11,  and  the  third  14.     How  many  scholars  were  study- 
ing arithmetic  ? 

13.  A  merchant,  on  closing  liis  business  for  the  day, 
found  he  had  received  23  dolldis  from  one  customer,  57 
from  another,  31  from  another,  and  25  from  various  oth- 
ers.    How  much  did  he  receive  that  day  ? 

14.  A  laborer,  in  pursuit  of  employment,  walked  7  miles 
the  first  day,  10  the  second,  12  the  third,  15  the  fourth, 
and  20  the  fifth  day.     How  far  had  he  then  walked  ? 

15.  A  man,  owning  a  large  farm,  gave  to  one  of  hia 
sons  112  acres,  to  another  123,  to  the  third  147,  and  had 
200  acres  left.     How  large  was  his  farm  at  first  ? 

16.  A  man  bought  a  barrel  of  oil  for  30  dollars,  and  sold 
it  so  as  to  gain  15  dollars.     How  much  did  he  sell  it  for  1 

17.  A  lad  bought  a  geography  for  50  cents,  a  grammar 
for  25  cents,  an  arithmetic  for  13  cents,  and  a  slate  for  10 
cents.     How  much  did  he  give  for  them  all  ? 

18.  A  gentleman  purchased  a  carpet  for  38  dollars,  a 
dozen  chairs  for  36  dollars,  a  bureau  for  15  dollars,  and  a 
table  for  12  dollars.     What  did  his  bill  amount  to  ? 

19.  A  merchant  had  4  notes;  one  for  157  dollars,  an- 
other for  368,  another  for  576,  and  another  for  1687  dol- 
lars.    What  was  the  whole  amount  of  his  notes  ? 

20.  A  gentleman  bought  a  cloak  for  56  dollars,  a  coat 
for  25  dollars,  a  vest  for  9  dollars,  a  hat  for  7  dollars,  and  a 
pair  of  boots  for  5  dollars.    What  did  he  give  for  the  whole  1 

21.  A  fashionable  lady   purchased  a  cashmere  shaw 
for  469  dollars,  a  watch  for  237  dollars,  a  pocket  hand 
kerchief  for  87  dollars,  and  a  bonnet  for  53  dollars.    Wha< 
was  the  amount  of  her  bill  ? 

22.  A  farmer  had  375  sheep  and  168  lambs  in  one  pas- 
ture, in  another  379  sheep  and  197  lambs.     How  many 
sheep  had  he  ?     How  many  lambs  ?     How  many  sheep 
and  lambs  together  ? 

23.  Four  men  entered  into  partnership  ;  one  furnished 
2878  dollars,  another  1784  dollars,  a  third  1265  dollars, 
and  the  fourth  894  dollars.    What  was  the  amount  oj 
their  sto*k? 


ART.  29.]  ADDITION.  37 

24.  A  man  sold  three  house  lots ;  for  one  he  received 
975  dollars,  for  another  763  dollars,  and  for  the  third  586 
dollars.     What  did  the  whole  amount  to  ? 

25.  A  gentleman  purchased  a  store  for  4500  dollars, 
and  paid  75  dollars  for  repairs,  and  150  dollars  for  having 
it  enlarged.     For  how  mucn  must  he  sell  it  in  order  to 
gain  175  dollars? 

26.  A  gentleman  paid  75  dollars  for  one  piece  of  cloth, 
67  dollars  for  another,  54  dollars  for  another,  and  48  dol- 
lars for  another.     How  much  did  he  pay  for  all  ? 

27.  A  certain  orchard  contains  56  apple-trees,  19  peach- 
trees,  23  plum-trees,  and    15  cherry-trees.     How  many 
trees  are  there  in  the  orchard  ? 

28.  The  distance  from  New  York  to  Albany  is  150 
miles,  from  Albany  to  Utica  93  miles,  from  Utica  to  Roch 
ester  158  miles,  and  from  Rochester  to  Buffalo  75  miles'. 
How  far  is  it  from  New  York  to  Buffalo  1 

29.  A  man  being  asked  his  age,  said  he  was  17  years 
old  when  he  left  the  academy,  he  spent  4  years  in  college, 
3  years  in  a  law  school,  practiced  law  15  years,  was  a 
member  of  congress  18  years,  and  it  was  16  years  since  he 
retired  from  business.     How  old  was  he  ? 

30.  A  shopkeeper  having  a  note  due,  paid  184  dollars 
at  one  time,  at  another  268  dollars,  at  another  379  dollars, 
at  another  467  dollars,  and  there  were  350  dollars  still 
unpaid.     What  was  the  amount  of  his  note  ? 

31.  A  gentleman  owns  a  house  and  lot  worth  10800 
dollars,  a  store  worth  5450  dollars,  a  house-lot    worth 
3700  dollars,  and  has  15000  dollars  in  personal  property, 
What  is  the  whole  amount  of  his  property  ? 

32.  A  man  left  his  estate  to  his  wife,  his  three  sons, 
and  two  daughters;  to  his  wife  he  gave  10350  dollars, 
to  his  sons  5450  dollars  apiece,  and  his  daughters  3500 
dollars  apiece.     How  large  was  his  estate  ? 

33.  A  merchant,  on  looking  over  his  accounts,  found 
he  owed  one  man  750  dollars,  another  648,  another  597, 
another  486,  another  379,  and  another  287  dollars.    What 
was  the  amount  of  his  debts  ? 

34.  A  man  bought  a  span  of  horses  for  275  dollars,  a 


38  ADDITION.  [SECT.  II 

carriage  for   150  dollars,  and  a  harness  for  87  dollars. 
How  much  did  he  give  for  the  whole  ? 

35.  A  man  bought  268  bushels  of  wheat  for  287  dol- 
lars, 187  bushels  of  corn  for  98  dollars,  and  .156  bushels 
of  oats  for  128  dollars.     How  many  bushels  of  grain  did 
he  buy ;  and  how  much  did  he  give  for  the  whole  ? 

36.  A  man  wishing  to  stock  his  farm,  paid  197  dollars 
for  a  span  of  horses,  86  dollars  for  a  yoke  of  oxen,  175 
dollars  for  cows,  and  169  dollars  for  sheep.     How  much 
did  he  give  for  the  whole  1 

37.  A  butcher  sold  to  one  customer  157  pounds  of  meat, 
to  another  159,  to  another  149,  to  another  97,  and  to  an- 
other 68.     How  much  did  he  sell  to  all  ? 

38.  A  carpenter  received  879  dollars  for  one  job,  for 
another  786,  for  another  693,  for  another  587,  for  another 
476,  and  for  another  368  dollars.     How  much  did  he  re- 
ceive in  all  ? 

39.  A  grocer  bought  375  dollars  worth  of  sugar,  287 
dollars  worth  of  molasses,  168  dollars  worth  of  tea.  158 
dollars  worth  of  coffee,  and  137  dollars  worth  of  spices. 
What  was  the  amount  of  his  bill  ? 

40.  A  merchant  bought  calico  to  the  amount  of  568 
dollars,  silks  to  the  amount  of  479  dollars,  and  broad- 
cloths to  the  amount  of  784  dollars.     He  sold  them  so  as 
to  gain  134  dollars  on  the  calico,  178  dollars  on  the  silks, 
and  242  dollars  on  the  broadcloths.     How  much  did  he 
sell  them  for  ;  and  what  was  the  amount  of  his  gains  ? 

41.  A  merchant  pays  560  dollars  a  year  for  store  rent, 
386  dollars  to  one  clerk,  267  to  another,  and  369  dollars 
for  various  other  expenses.     What  does  it  cost  him  a  year 
to  carry  on  his  business  ? 

42.  A  man  receives  568  dollars  rent  for  one  store,  479 
for  another,  and  276  for  another.     How  much  does  he  re- 
ceive  for  them  all  ? 

43.  The  distance  from  Boston  to  Springfield  is  98  miles, 
from  Springfield  to  Pittsfield  is  53  miles,  from  Pittsfield 
to  Albany  is  49  miles,  from  Albany  to  Auburn  is  173 
miles,  and  from  Auburn  to  Buffalo  is  152  miles.     Ho\r 
far  is  it  from  Boston  to  Buffalo  ? 


A.RT.  29.]  ADDITION.  39 

44.  A  n>9n  bought  a  quantity  of  oil  for  2649  dollars, 
and  candies  for  1367  dollars;  he  afterwards  sold  them  so 
as  to  ga;u  568  dollars  on  the  oil,  and  346  dollars  on  the 
candles.     How  much  did  he  receive  for  the  whole  ? 

45.  Ii  1840,  the  state  of  Maine  contained  501793  in- 
habitants ;  New  Hampshire,  284574 ;  Vermont,  291948  ; 
Massachusetts,  73/699;  Connecticut,  309978;  and  Rhode 
[sland,  103830.    *What  was  the  population  of  New  Eng- 
land? 

46.  In  1840,  the  state  of  New  York  contained  2428921 
inhabitants;  New  Jersey,  373306;  Pennsylvania,  1724- 
033  ;  and  Delaware,  78085.     What  was  the  population 
of  the  Middle  States? 

47.  In  1840,  the  state  of  Maryland  contained  470019* 
inhabitants  ;  Virginia,  1239797  ;  North  Carolina,  753419  ; 
South  Carolina,  594398;    Georgia,  691392;    Alabama, 
590756;    Mississippi,  375651;  and  Louisiana,  352411. 
What  was  the  population  of  the  Southern  States  ? 

48.  In  1840,  the  state  of  Tennessee  contained  829210 
inhabitants;  Kentucky,  779828  ;  Ohio,  1519467  ;  Michi- 
gan 212267;  Indiana,  685866;  Illinois,  476183;  Mis- 
souri, 383702  ;    and  Arkansas,  97574.     What  wdf  the 
population  of  the  Western  States  ? 

49.  In  1840,  the  territory  of  Florida  contained  54477 
inhabitants;  Wisconsin,  30945;  Iowa,  43112;    and  the 
District  of  Columbia,  43712  ;    on  board  vessels  of  war, 
6100.     What  was  the  population  of  the  Territories  and 
naval  service  of  the  United  States  ? 

50.  What  was  the  whole  population  of  the  United  States 
in  1840? 

*   iccording  to  Ine  Official  Revision. 


40  SUBTRACTION.  [SECT.  III. 

SECTION  III. 
SUBTRACTION. 

MENTAL   EXERCISES. 

ART.  3O.  Ex.  1.  Henry  having  7  peaches,  gave  4  to 
nis  sister  :  how  many  had  he  left  ? 

OBS.  To  solve  this  question,  consider  what  number  added  to  4 
makes  7.  Now  from  addition  we  know  that  4  and  3  make  7 ;  that 
is,  7  is  composed  of  the  numbers  4  and  3.  It  is  evident,  therefore,  if 
one  of  these  numbers  be  taken  from  7,  the  other  number  will  be  left. 
Hence,  4  peaches  from  7  peaches  leave  3  peaches.  Ans.  3  peaches. 

2.  James  had  7  cents,  and  spent  three  of  them :  how 
many  had  he  left  ? 

3.  «iack  has  6  marbles  :  how  many  more  must  ke  get 
to  make  10? 

4.  A  farmer  having  9  cows,  sold  5  of  them :   how 
many  had  he  left  1 

5.  A  pound  of  raisins  costs  1 1  cents,  and  a  pound  of 
sugar  8  cents :  what  is  the  difference  in  their  prices  ? 

6.  In  a  stage   coach  there  were  10  passengers,  6  of 
whom  got  out  at  a  hotel :  how  many  remained  in  the 
coach  ? 

7.  Dick  bought  a  knife  for  12  cents,  and  having  but  7 
cents  in  his  pocket,  agreed  to  pay  the  rest  to-morrow . 
how  much  does  he  owe  for  it  ? 

8.  John  gathered  8  quarts  of  chestnuts :    how  many 
more  must  he  gather  to  make  14  quarts? 

9.  The  cost  of  a  cap  is  13  shillings,  and  the  cost  of  a 
comforter  is  3  shillings :  what  is  the  difference  in  theii 
cost? 

10.  Susan  is   15  years  old,  and   Harriet  is  only  9: 
what  is  the  difference  in  their  ages  ? 


ART.  SO.] 


SUBTRACTION. 


41 


SUBTRACTION  TABLE. 


2  from 

3  from 

4  from 

5  from 

6  from 

7  from 

8  from 

9  from 

-2  lC£ 

ives  0 

3  lea.  0 

4  lea.  0 

5  lea.  0 

6  lea.  0 

7  lea.  0 

8  lea.  0 

9  lea.  0 

3 

'   1 

4 

1 

5  "  1 

6  "  1 

7  "  1 

8  "  1 

9 

1 

10  "  1 

4 

'   2 

5 

o 

6  "  2 

7  "  2 

8  "  2 

9 

2 

10 

2 

11  "  2 

5 

'   3 

6 

3 

7 

3 

8 

3 

9  "  3 

10 

3 

11 

3 

12 

3 

6 

'   4 

7 

4 

8 

4 

9 

4 

10  "  4 

11 

4 

12 

4 

13 

4 

7 

'   5 

8 

5 

9 

5 

10 

5 

11  "  5 

12 

5 

13 

5 

14 

5 

8 

1   6 

9 

6 

10 

6 

11 

6112  "  6 

13 

G 

14 

6 

15 

G 

9 

'   7 

10 

7 

11 

7 

12 

7 

13  "  7 

14 

7 

15 

7 

16 

7 

10 

'   8 

11 

8 

12 

8 

13 

8il4  "  8 

15 

8 

16  "  8 

17 

8 

11 

1   9 

19 

g 

13  "  9 

14 

9  15  "  9 

16 

9 

17  "  9 

18 

9 

12 

'  10  13 

10 

14  "  10 

15  "  10  16  "  10 

17  "  10  18  "  10 

19 

10 

OBS.  This  Table  is  the  reverse  of  Addition  Table.  Hence,  if  the 
pupil  has  thoroughly  learned  that,  it  will  cost  him  but  little  time  or 
trouble  to  learn  this.  (See  observations  under  Addition  Table.) 

11.  4  from  7  leaves  how  many?  4  from  9?  4  from 

12  ?  4  from  8  ?  4  from  1 1?  4  from  13  ? 

12.  6  from  8  leaves  how  many?  6  from   10?  6  from 

13  ?  6  from  1 1  ?  6  from  15  ?  6  from  12?  6  from  16  ? 
13.7  from  9  leaves  how  many  ?  7  from   1 1  ?  7  from 

14  ?  7  from  15  ?  7  from  16  ?  7  from  13  ?  7  from  17  ? 

14.  8  from  11?  8  from  13?  8  from  16?  8  from  12? 

8  from  15  ?  8  from  17  ?  8  from  14  ?  8  from  18  ? 

15.  9  from  12?  9  from  14?  9  from  11?  9  from  13? 

9  from  17  ?  9  from  15  ?  9  from  18  ?  9  from  19  ? 

16.  2  from  4  leaves  how  many?  2  from  14?  2  from 
24?  2  from  34?  2  from  44?  2  from  54?  2  from  64?  2 
from  74?  2  from  84?  2  from  94  ? 

17.  3  from  6  ?  3  from  16  ?  3  from  26  ?  3  from  36  ?  3 
from  46  ?  3  from  56  ?  3  from  66  ?  3  from  76  ?  3  from 
86  ?  3  from  96  ? 

18.  4  from  9  ?  4  from  29  ?  4  from  39  ?  4  from  49  ?  4 
from  59  ?  4  from  69  ?  4  from  79  ?  4  from  89  ?  4  from  99  ? 

19.  6  from  15  ?  6  from  25  ?  6  from  35  ?  6  from  45  ? 
6  from  55  ?  6  from  65  ?  6  from  75  ?  6  from  95  ? 

20.  8  from  14?  8  from  24?  8  from  34?  8  from  44?  8 
from  54  ?  8  from  64  ?  8  from  74  ?  8  from  84  ?  8  from  94  ? 

21.  A  gentleman  bought  a  coat  for  15  dollars,  and  a  hat 
for  6  dollars :  how  much  more  did  his  coat  cost  than  his 
hat? 


4Q  SUBTRACTION.  [SECT.  Ill, 

22.  A  farmer  having  sold  6  cords  of  wood  for  18  dol« 
lars,  took  a  barrel  of  flour  at  6  dollars  towards  his  pay 
and  the  rest  in  cash  :  how  much  money  did  he  receive  1 

23.  A  lady  bought  a  shawl  for  15  dollars,  and  hand 
ed  the  shopkeeper  a  20  dollar  bill :  how  much  change 
ought  she  to  receive  back  ? 

24.  A  man  having  25  watermelons  in  his  garden,  some 
wicked  boys  stole  9  of  them :  how  many  had  he  left  ? 

25.  James  is  14  years  old,  and  his  sister  is  19  :  what 
is  the  difference  in  their  ages? 

26.  A  merchant  had  a  piece  of  calico  which  contained 
33  yards  ;  on  measuring  the  remnant  he  finds  he  has  but 
7  yards  left:  how  many  yards  has  he  sold? 

27.  A  hogshead  of  cider  contains  63   gallons :  after 
drawing  out  9  gallons,  how  many  will  be  left  ? 

28.  Henry  had  48  silver  dollars,  and  gave  8  to  the  or- 
phan asylum :  how  many  dollars  did  he  have  left  ? 

29.  A  mim  bought  a  piece  of  cloth  containing  39  yards, 
and  sold  6  yards  of  it :  how  many  yards  had  he  left  ? 

30.  George  gave  75  cents  for  a  pair  of  skates,  and  sold 
them  for  9   cents  less  than  he  gave :  how  much  did  he 
get  for  his  skates  1 

31.  William  had   67  cents;  he  spent  5  for  chestnuts 
and  2  for  apples  :  how  many  cents  has  he  left  ? 

32.  A  man  sold  a  load  of  wood   for  18  shillings;  he 
laid  out  4  shillings  for  tea  and  6  for  sugar  :  how  many 
shillings  had  he  to  carry  home  ? 

33.  Sarah  having  85  cents,  gave  10  cents  to  the  Sab- 
bath School  Society,  8  to  the  Bible  Society,  and  spent  6 
for  candy :  how  many  cents  had  she  left  ? 

34.  If  I  pay  27  dollars  for  a  cow  and  sell  it  for  18  dol- 
lars, how  much  do  I  lose  by  the  bargain  ? 

35.  Richard  had  45  marbles ;  he  lost  7  and  gave  away 
5  :  how  many  had  he  left  ? 

36.  A  man  having  56  dollars  in  his  pocket,  bought  a 
hat  for  5  dollars,  a  coat  for  10,  and  a  pair  of  boots  for  4 
how  much  money  had  he  left? 

37.  If  I  owe  a  merchant  50  dollars  and  pay  him  20 
dollars,  how  many  dollars  shall  I  then  owe  him  ? 

Ans.  30  dollars. 


ART.  81.]  SUBTRACTION.  43 

Suggestion. — It  is  advisable  for  beginners  to  analyze 
the  numbers  in  this  question,  as  in  Art.  16,  Ex,  31,  and 
then  take  2  tens  from  5  tens. 

38.  A  farmer  having  80  sheep,  sold  all  but  30 :  how 
many  did  he  sell? 

39.  A  man  having  90  acres  of  land,  gave  50  acres  to 
his  son :  how  many  acres  has  he  left  ? 

40.  George  had  70  cents  and  spent  30 :  how  many 
had  he  left? 

41.  In  a  certain  orchard  there  are  100  trees,  60  of 
them  are  apple-trees  and  the  rest  are  peach-trees :  how 
many  peach-trees  are  there  ? 

42.  A  grocer  bought  150  eggs,  and  afterwards  found 
that  20  of  them  were  rotten :    how  many  sound  ones 
were  there  ? 

43.  In  the  Centre  School  there  are  150  scholars,  60 
of  whom  are  girls  :  how  many  boys  are  there  ? 

44.  A  man  bought  a  horse  for  90  dollars,  and  sold  it 
immediately  for  130  dollars  :  how  much  did  he  make  by 
his  bargain  ? 

45.  A  man  owing  me  200  dollars,  turned  me  out  a 
horse  worth  80  dollars,  and  is  to  pay  the  balance  in  cash : 
how  much  money  must  he  pay  me  ? 

46.  A  boy  going  to  market  with  80  cents,  bought  20 
cents  worth  of  cheese,  and  30  cents  worth  of  butter :  how 
much  change  had  he  left  ? 

47.  35  from  42  leaves  how  many?  63  from  75? 

48.  26  from  40  leaves  how  many  ?  35  from  45  ? 

49.  65  from  85,  how  many  ?  82  from  94,  how  many  ? 

50.  8  from  17,  how  many?   13  from  26,  how  many? 
6  from  25,  how  many  ?  8  from  94,  how  many  ?  5  from 
68,  how  many?   17  from  34,  how  many?    7  from  43, 
how  many?  6  from  72,  how  many?  9  from   75,  how 
many  ?  7  from  86,  how  many  ? 

3 1  •  It  will  be  observed  that  all  the  preceding  exam 
pies  of  this  section,  though  expressed  in  a  variety  of 
ways,  involve  the  same  principle ;  that  the  object  aimed 
at  in  each  of  them,  is  to  find  the  difference  between  two 
numbers;  consequently,  they  are  all  performed  in  the 


44  SUBTRACTION.  [SECT.  1IL 

same  manner.     The  operation  consists  in  taking  a  les\ 
number  from  a  greater,  and  is  called  subtraction.     Hence, 

32S.  SUBTRACTION  is  the  process  of  finding  the  differ- 
ence between  two  numbers. 

The  difference,  or  the  answer  to  the  question,  is  called 
the  remainder. 

OBS.  1.  The  number  to  be  subtracted  is  often  called  the  subtraherd, 
and  the  number  from  which  it  is  subtracted,  the  minuend.  These 
terms,  however,  are  calculated  to  embarrass,  rather  than  assist  the 
earner,  and  are  properly  falling  into  disuse. 

2.  Subtraction,  it  will  be  perceived,  is  the  reverse  of  addition.     Ad 
dition  unites  two  or  more  numbers  into  one  single  number ;  subtrac 
tion,  on  the  other  hand,  separates  a  number  into  two  parts. 

3.  When  the  given  numbers  are  of  the  same  denomination,  the 
operation  is  called  Simple  Subtraction.     (Art.  18.  Obs.) 

33.  Subtraction  is  often  represented  by  a  short  hori- 
zontal line  ( — ),  which  is  called  minus.     When  placed  be 
tween  two   numbers,  this  sign  shows  that  the  number 
after  it  is  to  be  subtracted  from  the  one  before  it.     Thus 
the  expression-  8 — 5,  signifies  that  5  is  to  be  subtracted 
from  8  ;  and  is  read,  "  8  minus  5,"  or  "  8  less  5." 

Note. — The  term  minus. is  a  Latin  word  signifying  less. 

EXERCISES    FOR    THE    SLATE. 

34.  When  we  wish  to  find  the  difference  between 
two  small  numbers,  it  is  the  most  convenient  way  to  per- 
form the  subtraction  in  the  mind.     Bat  when  the  num- 
bers are  large,  it  is  difficult  to  retain  them  in  the  mind, 
and  carry  on  the  operation  at  the  same  time.     By  setting 
them  down  upon  a  slate  or  black-board,  however,  the 
process  of  subtracting  large  numbers  is  rendered  short 
and  simple.   (Art.  21.) 


Q. — What  is  subtraction?  What  is  the  answer  called  ?     Obs.  What 

is  the  number  to  be  subtracted  sometimes  called?     That  from  which 

it  is  sublrnct/.'d  ?    Of  wiiat  is  subiraction  the  reverse  ?    When  the  given 

uumhrrs  are  of  the  same  denomination,  what  is  the  operation  called  ? 

33.  Wh:it  is  the  si#7i  of  subtraction  railed  ?     Of  what  does  it  consist ' 

What  does  it  show?     How  is  ihe expression  8 — 5,  read?     Note.  What 

in:;'  of  til1,'  term  minus  ?     34.  What  is  the  most  convenient 

of  finding  the,  difference  between  two  small  numbers?     WhrU 

!  two  large  ones  ? 


ARTS.  32-*84.]  SUBTRACTION.  45 

Ex.  1.  Suppose  a  man  gave  475  dollais  for  a  span  oi 
horses,  and  352  dollars  for  a  carriage :  how  much  more 
did  he  pay  for  his  horses  than  for  his  carriage  ? 

Directions. — Write    the    less  Operation. 

number   under   the   greater,    so  ^      M 

that  units  may  stand  under  units, 
tens  under  tens,  &c.     Now,  be- 
ginning with  the  units,  proceed     Horses,    475  Dolls, 
thus :  2  units  from  5  units  leave     Carriage,  3  5  2  Dolls. 
3  units;   write  the   3  in   units'     Rem.        1  2  3  Dolls, 
place,  under  the  figure  subtract- 
ed.    5  tens  from  7  tens  leave  2  tens ;  set  the  2  in  tens' 
place.     3  hundreds  from  4  hundreds  leave  1  hundred ; 
write  the  1  in  hundreds'  place.     The  remainder  is  123 
dollars. 

OBS.  It  is  important  for  the  learner  to  observe,  that  we  subtract 
units  from  units,  tens  from  tens,  &c. ;  that  is,  we  subtract  figures  of 
the  same  order  from  each  other.  This  is  done  for  the  same  reason 
that  we  add  figures  of  the  same  order  to  each  other.  (Art.  22.) 
Hence,  in  writing  numbers  for  subtraction,  great  care  should  be  taken 
to  set  units  under  units,  &c.,  in  order  to  prevent  the  mistake  of  sub- 
tracting different  orders  from  each  other. 

2.  A  merchant  bought  268  barrels  of  flour ;  and  on  ex- 
amination, found  that  only  123  barrels  were  fit  for  use" 
how  many  were  damaged?    Ans.  145. 

Suggestion. — Write  the  less  number  under  the  greater, 
&c.,  and  proceed  as  above. 

3.  A  traveler  having  576  dollars,  was  robbed  of  344 
dollars :  how  many  dollars  had  he  left  ? 

4.  What  is  the  difference  between  648  and  235  ? 

5.  What  is  the  difference  between  876  and  523  ? 

6.  What  is  the  difference  between  759  and  341  ? 

7.  What  is  the  difference  between  4567  and  1235? 

8.  What  is  the  difference  between  8643  and  5412  ? 


QUEST. — In  the  1st  example  how  do  you  write  the  numbers  for  sub- 
traction ?  Where  begin  to  subtract  ?  Obs.  What  orders  do  you  sub- 
tract from  each-  other  ?  Why  not  subtract  different  orders  from  each 
other  1  Why  place  units  under  units,  &o.,  in  subtraction  ? 


46  SUBTRACTION.  [SECT.  Ill 

9.  10.  11.  12. 

From  68476  765274  563181         3286732 

Take   36124  152140  32040  135011 

35.  When  the  figures  in  the  lower  number  are  all 
smaller  than  those  directly  over  them,  each  lower  figure, 
as  we  have  seen  in  the  preceding  examples,  must  be  sub- 
tracted from  that  above  it,  and  the  remainder  must  be 
placed  under  the  figure  subtracted. 

But  it  often  happens  that  a  figure  in  the  lower  numbei 
is  larger  than  that  above  it,  and  consequently  cannot  be 
taken  from  it. 

13.  It  is  required  to  find  the  difference  between  75 
and  48. 

It  is  plain  that  we  cannot  take  8  units     Operation. 
from  5  units,  for  8  is  larger  than  5.     What  75 

then  shall  we  do  ?     Since  75  is  composed  43 

of  7  tens  and  5  units,  we  can  take   1  ten  -^=  „ 

from  the  7  tens,  and  adding  it  mentally  to 
the  5  units,  it  will  make  15  units.  Then  subtracting  the 
8  units  from  15  units,  will  leave  7  units ;  write  the  7  un- 
der the  units'  column.  As  we  took  1  ten  from  the  7  tens, 
we  have  but  6  tens  left ;  and  4  tens  from  6  tens  leave  2 
tens :  write  the  2  under  the  tens'  column.  The  whole 
remainder,  therefore,  is  2  tens  and  7  units,  01  27. 

36.  The  process  of  taking  one  from  a  higher  order 
in  the  upper  number,  and  adding  it  to  the  figure  froxsi 
which  the  subtraction  is  to  be  made,  is  called  borrowing 
ten,  and  is  the  reverse  of  carrying  ten.    (Art.  24.) 

OBS.  The  1  taken  from  a  higher  order,  is  always  equal  to  10  ID 
the  next  lower  order  to  which  it  is  added.  (Art.  8.) 

37*  The  principle  of  borrowing  may  be  illustrated 
by  the  following  analytic  solution  of  the  last  example. 


QUEST. — 35.  When  the  figures  in  the  lower  number  are  each  small 
er  than  those  over  them,  how  proceed  ?  Where  do  you  place  the  re- 
mainder ?  Is  a  figure  in  the  lower  number  ever  larger  than  that  above 
it  ?  36.  What  is  meant  by  borrowing  10  ?  What  is  the  1  taken  from  the 
nighr-r  order  equal  to  ? 


A.RTS.  35-37,  a.]         SUBTRACTION.  47 

75=604-15  Taking  1  ten  from  7  tens,  and 

uniting   it   with    the  5  units,    we 


7,  or  27.  number.  And  we  simply  separate 
the  lower  number  into  the  tens  and  units  of  which  it  is 
composed.  Now  subtracting,  as  in  the  last  article,  8  from 
15  leaves  7  :  40  from  60  leaves  20.  Thus  the  remain- 
der is  20-f  7,  or  27,  the  same  as  before. 

OBS.  It  is  manifest  that  this  process  of  borrowing  ten,  does  not 
change  the  value  of  the  upper  number  ;  for,  it  consists  simply  in 
transposing  a  part  of  one  order  to  another  order  in  the  same  number, 
which  can  no  more  diminish  or  increase  the  number,  than  it  will  di- 
minish or  increase  the  amount  of  money  a  man  has,  if  he  takes  a 
part  from  one  pocket  and  puts  it  into  another.  It  is  advisable  for  the 
pupil  to  analyze  several  examples  as  above,  until  the  process,  of  bor- 
rowing becomes  familiar. 

.  A    ^  Since  7  units  cannot  be  taken  from 

2  units>  we  borrow  10,  which  added 


_  to  the  2.  will  make  12  :  then  7  units 
Bern,  3675  frorn  12  units  leave  5.  Now  hav- 
ing borrowed  1  of  the  4  tens,  it  becomes  3  tens  ;  and  6 
from  3  is  impossible  :  hence  we  must  borrow  again.  But 
the  next  figure  in  the  upper  number,  i.  e.  the  figure  in  the 
hundreds'  place,  is  a  0,  and  consequently  has  nothing  to 
lend.  We  must  therefore  borrow  1  from  the  next  order 
still,  i.  e.  from  thousands,  and  adding  it  to  the  0,  it  will 
make  10  hundreds.  Then,  borrowing  1  of  the  10  hun 
dreds  and  adding  it  to  the  3  tens,  it  will  make  13  tens, 
and  6  from  13  leaves  7.  Diminishing  the  10  hundreds 
oy  1,  (which  we  borrowed,)  it  becomes  9,  and  3  from  9 
leaves  6.  Again,  diminishing  the  6  thousands  by  1, 
(which  we  borrowed,)  it  becomes  5,  and  2  from  5  leaves 
3.  The  answer  is  3675. 

3  7  .  a.  There  is  another  method  of  borrowing,  or  rath- 
er  of  paying,  which  the  learner  will  often  find  more  con- 


QUKBT. — How  illustrate  the  principle  of  borrowing  upon  the  black- 
board ?  Obs.  Is  the  value  of  the  upper  number  increased  by  borrow- 
ing? Is  it  diminished?  How  does  this  appear?  37.  a.  When  w« 
borrow  10,  what  other  way  is  there  to  compensate  for  it ! 


48  SUBTRACTION.  [SECT,  HI 

venient  In  practice  than  the  preceding-,  and  less  liable  t« 
lead  him  into  mistakes,  especially,  when  the  figure  in  the 
next  higher  order  is  a  cipher. 

When  we  borrow  10,  that  is,  when  we  add  10  to  the 
upper  figure,  instead  of  considering  the  next  figure  in  the 
upper  number  to  be  diminished  by  1,  the  result  will  mani- 
festly be  the  same,  if  we  simply  add  1  to  the  next  figure 
in  the  lower  number. 

Thus,  in  the  last  example,  instead  of  diminishing  the  4 
tens  in  the  upper  number  by  1,  we  may  add  1  to  the  6 
tens  in  the  lower  number,  which  will  make  7 ;  and  7  from 
14  leaves  7,  the  same  as  6  from  13.  Again,  adding  1  to 
the  3  hundreds  (to  compensate  for  the  10  we  borrowed) 
makes  4  hundreds  ;  and  4  from  10  leaves  6,  the  same  as  3 
from  9.  Finally,  adding  1  to  the  2  (because  we  borrowed) 
makes  3 ;  and  3  from  6  leaves  3.  The  remainder  is  3675, 
the  same  as  before. 

*7/t         6  from  4  is  impossible :  add  10  to 
3^6     th«  4,  and  it  will  make   14;  then  6 
from  14  leaves  8.     Adding  1  to  the  2 
n       OAo     makes  3,  and  3  from  7  leaves  4.     3 

from  5  leaves  2.     Ans.  248. 

Ozs.  This  method  of  borrowing  depends  on  the  self-evident  prin- 
ciple, that  if  any  two  numbers  are  equally  increased,  their  difference 
will  not  be  altered.  That  the  two  given  numbers  are  equally  in- 
creased by  this  process,  is  evident  from  the  fact  that  the  1  added  to 
the  lower  number,  is  of  the  next  superior  order  to  the  10  added  to  the 
upper  number,  and  will  compensate  for  it ;  for  1  in  a  superior  order, 
is  equal  to  10  in  an  inferior  order.  (Art.  8.)  Hence, 

38.  When  a  figure  in  the  lower  number  is  larger 
than  that  above  it,  borrow  10,  i.  e.  add  10  to  the  upper 
figure,  and  from  the  number  thus  produced,  subtract  the 
lower  figure  :  to  compensate  this,  add  1  to  the  next  figure 
in  the  lower  number ;  or  diminish  the  next  figure  in  the 
upper  number  by  1,  and  proceed  as  before. 

16.  17.  18. 

From    78562  645630  70430256 

Take     24380  520723  4326107 

QUEST.— -Obs.  Upon  what  does  the  second  method  of  borrowing  de- 
pend ?  How  does  it  appear  that  you  increase  the  given  number* 
equally  ? 


ARTS.  38-40.]  SUBTRACTION.  49 


39.    PROOF.  —  Add  the  remainder  to  the.  smaller 
ber  ;  and  if  the  sum  is  equal  to  the  larger  number,  the  work 
is  right. 

19.  A  man  bought  a  horse  for  175  dollars,  and  sold  it 
for  127  dollars:  how  much  did  he  lose  by  his  bargain? 

Operation.  Proof.  Since  the  sum  of  the 

Paid      1  75  dolls.      1  27  Smaller  No.      im?11f  number  and  rc- 
Rec'd   127  dolls.         8  Remainder. 


Lost       48  dolls.      175  Larger  No.      ration  is  correct. 

OBS.  This  method  of  proof  depends  upon  the  obvious  principle, 
that  if  the  difference  between  two  numbers  be  added  to  the  less,  the 
sum  must  be  equal  to  the  greater. 

20.  From  8796  subtract  2675,  and  prove  the  operation  ? 

21.  From  6210896  subtract  3456809,  and  prove  the 
operation. 

22.  From  1000000  subtract  67583,  and  prove  the  ope- 
ration. 

23.  From  7834501  subtract   1000000,  and  prove  the 
operation. 

24.  From  68436907  subtract  59476012,  and  prove  the 
operation. 

25.  From  8006754231  subtract  79756634  17,  and  prove 
the  operation. 

4O.  From  the  preceding  illustrations  and  principles 
we  derive  the  following 

GENERAL  RULE  FOR  SUBTRACTION. 

I.  Write  the  kss  number  under  the  greater,  so  that  untt* 
may  stand  under  units,  tens  under  tens,  &c. 

II.  Beginning  at  the  right  hand,  subtract  each  figure  in 
(he,  lower  number  from  thefiguie  above  it,  and  set  the  remain- 
der directly  under  the  figure  subtracted.  (Art  35.) 


QUEST. — 38.  How  then  do  you  proceed,  when  a  figure  in  the  lower 
number  is  larger  than  the  one  over  it  I  Why  do  you  add  1  to  the  next 
figure  in  the  lower  line  ?  39.  How  is  subtraction  proved  ?  Obs.  Up- 
on v>  hat  principle  does  the  proof  of  subtraction  depend  ?  40.  What  ia 
the  general  rule  for  subtraction;  1 


50  SUBTRACTION.  [SECT.  Ill 

III.  Whe?i  a  figure  in  the  lower  number  is  larger  than 
that  above  itj  add  10  to  the  upper  figure;  then  subtract  a% 
before,  and  add  1  to  the  next  figure  in  the  lower  number. 
(Arts.  37,  38.) 

EXAMPLES   FOR   PRACTICE. 

1.  A  man  bought  a  piece  of  cloth  containing  37  yards, 
and  sold  24  yards  of  it.     How  much  had  he  left  ? 

2.  A  merchant  had  on  hand  a  quantity  of  flour,  for 
which  he  asked  245  dollars;  but  for  ready  money  he 
made  a  deduction  of  24  dollars.     How  much  did  he  re- 
ceive for  his  flour  ? 

3.  In  a  certain  Academy  there  were  357  scholars,  168 
of  whom  were  young  ladies.     How  many  young  gentle- 
men were  there  1 

4.  A  farmer  raised  4879  bushels  of  wheat,  and  sold 
3876  bushels.     How  much  had  he  left  ? 

5.  A  man  purchased  a  farm  for  4687  dollars,  but  the 
times  becoming  hard  he  was  obliged  to  sell  it  for  896 
dollars  less  than  he  gave  for  it.     How  much  did  he  sell 
it  for? 

6.  A  merchant  bought  2268  dollars  worth  of  goods, 
which,  in  consequence  of  getting  damaged,  he  sold  for 
848  dollars  less  than  cost.     How  much  did  he  sell  them 
for? 

7.  A  merchant  sold  a  lot  of  silks  for  561  dollars,  which 
was  179  dollars  more  than  the  cost  of  them.     How  much 
did  he  give  for  them  ? 

8.  A  man  bought  an  estate  for  8796  dollars,  and  sold 
it  again  for  9875  dollars.     How  much  did  he  gain  by  his 
bargain  ? 

9.  A  farmer  raised  1389  bushels  of  wheat  one  year, 
and  1763  the  next.     How  much  more  did  he  raise  the 
second  year  than  the  first  ? 

10.  A  man  bought  a  house  and  lot  for  5687  dollars 
The  house  was  worth  3698  dollars .-  how  much  was  the 
lot  worth? 

11.  Suppose  a  gentleman's  income  is  3268  dollars  a 
year,   and  his  expenses  are  2789  dollars.     How  much 
does  ne  save  in  a  year  ? 


\RT.  40.]  SUBTRACTION.  51 

12.  The  United  States  declared  their  independence  in 
1776 :  how  many  years  is  it  since  ? 

13.  Two  brothers  commenced  business  at  the  same 
time ;  one  gained  3678  dollars  in  five  years,  the  other 
gained  2387  dollars  in  the  same  time.     How  much  more 
lid  one  gain  than  the  other? 

14.  The  distance   from  Boston  to  Springfield   is   98 
miles,  and  from  Boston  to  Pittsfieldit  is  151  miles.   How 
far  is  it  from  Springfield  to  Pittsfield  ? 

15.  From  New  York  to  Utica  it  is  243  miles,  and  from 
New  York  to  Albany  it  is   150  miles.     How  far  is  it 
from  Albany  to  Utica'? 

16.  America  was  discovered  by  Columbus  in   1492: 
how  many  years  is  it  since? 

17.  Dr.  Franklin  died  A.  D.  1790,  and  was  84  years 
old  when  he  died  :  in  what  year  was  he  born  ? 

18.  General   Washington  was  born  A.  D.   1732,  and 
died  in  1799  :  how  old  was  he  when  he  died? 

19.  The  first  settlement  in  New  England  was  made  at 
Plymouth  in  the  year  1620  :  how  many  years  is  it  since? 

20.  A  ship  sailed  having  on  board  a  cargo  valued  ai 
100000  dollars,  but  being  overtaken  by  a  storm,  27680 
dollars  worth  of  goods  were  thrown  overboard.     How 
much  of  the  cargo  was  saved  ? 

21.  The  population  of  Massachusetts  in   1840,   was 
737699,  and  that   of  Connecticut  was   309978.      How 
many  more  inhabitants  were  there  in  Massachusetts  than 
in  Connecticut? 

22.  In   1840.   the    population   of  Massachusetts    was 
737699,  and  in  '1820  it  was  523287.     How  much  did  the 
population  increase  during  this  period  ? 

23.  In  1840,  the  population  of  the  state  of  New  York 
was  2428921,  and  in  1820  it  was  1372812.     How  much 
did  the  population  increase  during  that  period  ? 

24.  In    1840,   the   population  of  the   New   England 
States  was  2234822,  and  that  of  the  State  of  New  York 
was  242S921.     How  many  more  inhabitants  were  there 
in  the  State  of  New  York  than  in  New  England? 

25.  In  1800,  the  population  of  the  United  States  was 
5305925,  and  in  1840  it  was  17069453.     How  much  d^d 
it  increase  irs  forty  year's?. 


52  SUBTRACTION.  [SECT.  Ill, 

26.  A  farmer  having  389  acres  of  land,  sold  to  onft 
man  126  acres,  and  to  another   163.     How  many  acres 
had  he  left  ? 

27.  A  gentleman  having  1768  dollars  deposited  in  the 
bank,  gave  a  check  for  175  dollars  to  one  man,  to  another 
for  238  dollars,  and  to  another  for  369  dollars.     How 
much  remained  on  deposit  ? 

28.  A  man  bought  a  horse  for  87  dollars,  a  carriage 
for  75  dollars,  and  a  harness  for  16  dollars,  and  sold  them 
all  together  for  200  dollars.     How  much  did  he  gain  by 
the  bargain  ? 

29.  A  man  bought  a  quantity  of  sugar  for  25  dollars,  a 
quantity  of  molasses  for  27  dollars,  and  a  quantity  of  rai- 
sins for  29  dollars,  for  which  he  paid  a  hundred  dollar- 
bill.     How  much  change  ought  he  to  receive  back  ? 

30.  An  orchard  contained  120  apple-trees,  47  peach- 
trees,  and  28  pear-trees.     Of  the  apple-trees  26  were  cut 
down  for  a  Railroad  to  pass  through,  18  of  the  peach- 
trees  died,  and  5    of  the  pear-trees  were   blown  down. 
How  many  trees  were  left  in  the  orchard  ? 

31.  A  gentleman  had  2700  dollars  which  he  wished  to 
distribute  among  his  three  sons.     To  the  oldest  he  gave 
825  dollars,  to  the  second  785  dollars,  and  the  remain- 
der to  the  youngest.     How  much  did  the  youngest  son 
receive  ? 

32.  A  man  owing  5648  dollars,  paid  at  one  time  536 
dollars,  at  another  378  dollars,  and  at  another  896  dollars. 
How  much  did  he  then  owe  ? 

33.  A  man  having  7689  dollars,  invested  689  dollars 
in  Railroad  stock,  500  dollars  in  a  woolen  factory,  and 
1250  dollars  in  bank  stock.     How  much  had  he  left  1 

34.  A  man  bought  a  quantity  of  oil  for  1763  dollars, 
and  a  lot  of  candles  for  598  dollars.     He  afterwards  sold 
them  both  for  2684  dollars.     How  much  did  he  gain  by 
the  bargain? 

35.  A  man  owning  3789  acres  of  land,  gave  to  one 
son  869  acres,  and  to  another  987  acres.     How  much 
land  had  he  left  ? 

36.  A  ship  of  war  sailing  with  650  men,  lost  in  one 
oattle  29  men,  in  another  37,  and  by  sickness  19  more, 
How  many  were  still  living  1 


ART.  40.]  SUBTRACTION.  53 

37.  A  merchant  owes  one  man  2684  dollars,  another 
1786  dollars,  another  987  dollars.     The  whole  amount 
of  his  property  is  4684  dollars.     How  much  more  does 
he  owe  than  he  is  worth  ? 

38.  A  man  bought  three  farms:  for  the  first  he  gave 
4673  dollars,  for  the  second  5674  dollars,  and  for  the  third 
9287  dollars,     He  sold  them  all  for  37687  dollars.     How 
much  did  he  gain  by  the  bargain  ? 

39.  A  man  bought  86  dollars  worth  of  wheat,  48  dol- 
lars worth  of  butter,  and  a  fine  horse  worth  148  dollars. 
He  gave  his  note  for  128  dollars,  and  paid   the  rest  in 
:ash.     How  much  money  did  he  pay  ? 

40.  A  gentleman  left  a  fortune  of  18864  dollars  to  be 
divided  between  his  two  sons  and  one  daughter  ;  to  one 
son  he  gave  6389  dollars,  to  the  other  6984  dollars.    How 
much  did  the  daughter  receive  ? 

41.  A  man  owing  8648  dollars,  paid  at  one  time  486, 
at  another  684,  at  another  729  dollars.     How  much  did 
he  still  owe  ? 

42.  Suppose  a  man  gains  by  one  speculation  867  dol- 
lars, by  another  687;  another  time  he  gains  563  dollars, 
and  then  loses  479  ;  still  another  time  he  gains  435  dol- 
lars, and  loses  378.     How  much  more  has  he  gained  than 
lost? 

43.  A  man  borrowed  of  a  friend  684  dollars  at  one 
time,  786  at  another,  874  at  another,  and  976  at  another. 
He  has  paid  568  dollars.     How  much  does  he  still  owe  ? 

44.  If  a  man's  income  is  4586  dollars  a  year,  and  he 
spends  384  dollars  for  clothing,  568  for  house  rent,  784 
for   provisions,  568  for  servants,  and  369  for  traveling, 
how  much  will  he  have  left  at  the  end  of  the  year  ? 

45.  A  merchant  bought  a  quantity  of  sugar  for  8978 
dollars,  paid  374  dollars  freight,  and  then  sold  it  for  9684 
dollars.     How  much  did  he  gain  by  the  trade? 

46.  A  merchant  had  in  his  storehouse  6384  bushels 
of  wheat,  3752  bushels  of  corn,   4564  bushels  of  oats, 
and  1384  bushelp  of  rye  :  it  was  broken  open  and  3564 
bushels  of  grain   taken  out.     How   many  bushels    re- 
r  lained  ? 

47.  A  man  bought  a  quantity  of  beef  for  5493  dollars. 


54  MULTIPLICATION.  [SECT.  IV 

a  quantity  of  coffee  for  261  dollars,  and  a  quantity  of  su 
gar  for   157  dollars  ;  in  exchange  he  gave  3687  dolla 
worth  of  flour,  568  dollars  worth  of  oats,  and  165  dolla 
worth  of  potatoes.     How  much  did  he  then  owe  1 

48.  A  gentleman  has  real  estate  valued  at  3879  dol 
iars,  and  personal  property  amounting  to  9857  dollars, 
He  owes  one  man  1350  dollars,  and  another  2687  dollars. 
How  much  would  he  have  left  if  he  should  pay  his  debts? 

49.  A  man  having  property  amounting  to  30000  dollars, 
lost  by  fire  a  store  worth  5000  dollars,  and  goods  to  the 
amount  of  3578  dollars.     How  much  property  had  he 
left? 

50.  A  man  died  leaving  an  estate  of  175000  dollars. 
He  gave   to  his  wife   25000  dollars,  to   his  three  sons 
32000  apiece,  to  his  two  daughters,  23000  dollars  each, 
and  the  rest  he  gave  to  a  literary  institution.     How  much 
&d  the  institution  receive  ? 


SECTION  IV. 
MULTIPLICATION. 

MENTAL   EXERCISES. 

ART.  41*  Ex.  1.  What  will  3  lead  pencils  cost,  at  4 
cents  apiece  ? 

Solution. — Three  pencils  will  evidently  cost  three  times 
as  much  as  one  pencil.  Now  if  1  pencil  costs  4  cents,  3 
pencils  will  cost  3  times  4  cents ;  and  3  times  4  cents  are 
12  cents.  Ans.  12  cents. 

Note.— It  is  highly  important  for  the  pupil  to  give  the  reason  in 
full  for  the  solution  of  every  example. 

2.  What  will  2  yards  of  cloth  cost,  at  8  dollars  a  yard  1 

3.  At  6  cents  apiece,  what  will  4  oranges  cost  ? 

4.  What  cost  5  pounds  of  ginger,  at  7  cents  a  pound? 


ART.  41.] 


MULTIPLICATION. 


B» 


5.  If  1  pair  of  gloves  cost  6  shillings,  what  will  6  pair 
cost? 

6.  At  9  cents  a  pound,  what  will  4  pounds  of  butter 
come  to  ? 

7.  What  wi  II  7  barrels  of  flour  cost,  at  4  dollars  a  barrel  ? 

8.  In  1  bughel  there  are  4  pecks  :  how  many  pecks  are 
there  in  6  bushels  ? 

9.  What  cost  8  pair  of  boots,  at  6  dollars  a  pair  ? 

10.  At  9  shillings  apiece,  what  will  5  caps  cost? 

11.  What  cost  6  pounds  of  sugar,  at  10  cents  a  pound? 

12.  What  cost  9  inkstands,  at  8  cents  apiece? 

MULTIPLICATION  TABLE. 


2  times 

3  times 

4  times 

5  times 

6  times 

7  times 

1   are    2 

1    are    3 

1   are    4 

1    are    5 

1    are    6 

1   are    7 

•2 

4 

2 

6 

2    "     8 

2 

10 

2 

12 

2     "    14 

3 

6 

3 

9 

3     "    12 

3 

15 

3 

18 

3     "    21 

4 

8 

4 

12 

4 

16 

4 

20 

4 

24 

4 

28 

5 

10 

5 

15 

5 

20 

5 

25 

5 

30 

5 

35 

6 

12 

6 

18 

6 

24 

6 

30 

6 

36 

6 

421 

7 

14 

7 

21 

7 

28 

7 

35 

7 

42 

7 

49 

8 

16 

8 

24 

8 

32 

40 

8 

48 

8 

56 

9 
10 

18 
20 

9 
10 

27 
30 

9 
10 

36 
40 

9 
10 

45 

50 

9 
10 

54 

60 

9 

10 

63 
70 

11 

22 

11 

33 

11 

44 

11 

55J11 

66|11 

77 

13 

24 

12 

36 

12 

48 

12 

60  12 

72il2 

84 

8  times 

9  times 

10  times 

11  times     |     12  times 

I    are    8 

I    are    9 

I    are  10 

1    are  11 

1    are  12 

2 

1,6 

2    "     18 

2 

20 

2 

22 

2 

24 

3 

24 

'    27 

3 

30 

3 

33 

3 

36 

4 

32 

4 

'    36 

4 

40 

4 

44 

4 

48 

5 

40       5 

(    45 

5 

50 

5 

55 

5 

60 

6 

48  !     6 

'    54 

6 

60 

6 

66 

6 

72 

7 

56 

7 

63 

7 

70 

7 

77 

7 

84 

8 

64 

8 

72 

8 

80 

8 

88 

8 

96  i 

9 

72 

9 

81 

9 

90 

9 

99 

9 

108 

10 

80 

10 

90 

10 

100 

10 

110  I  10 

120 

11 

88 

11 

99 

11 

110 

11 

121  i  11 

132 

,  12 

90 

12 

108 

12 

120 

12 

132 

12 

144 

OBS.  The  pupil  will  find  assistance  in  learning  this  table,  by  ob- 
serving the  following  particulars. 

1.  The  several  results  of  multiplying  by  10  are  formed  by  simply 
adding  a  cipher  to  the  figure  that  is  to  be  multiplied.  Thus,  10  times 
2  are  20-  10  times  3  are  30,  &c. 


60*  MULTIPLICATION.  [SECT.  IV 

2.  The  results  of  multiplying  by  5,  terminate  in  5  and  0,  alter- 
nately.    Thus,  5  times  1  are  5 ;  5  times  2  are  10 ;  5  times  3  are  1 5,  &xx 

3.  The  first  nine  results  of  multiplying  by  11  are  formed  by  re> 
pcating  the  figure  to  be  multiplied.     Thus,  11  times  2  are  22;  11 
times  3  are  33,  <£c. 

4.  In  the  successive  results  of  multiplying  by  9,  the  right  hand 
figure  regularly  decreases  by  1,  and  the  left  hand  figure  regularly  in- 
creases by  1.     Thus,  9  times  2  are  18;  9  times  3  are  27;  9  times  4 
are  36,  &c. 

13.  At  2  dollars  a  cord,  what  will  12  cords  of  wood 
cost?   10  cords?  9  cords?  8  cords?  7  cords?  6  cords?  5 
cords  ?  4  cords  ?  3  cords  ? 

14.  In  one  yard  there  are  3  feet :  how  many  feet  are 
therein  12  yards?  in  11  yards?   10  yards?  9  yards?  8 
yards  ?  7  yards  ?  6  yards  ?  5  yards  ?  4  yards  ? 

15.  In  1  gallon  there  are  4  quarts:  how  many  quarts 
in  12  gallons?  in  11  gallons?   10  gallons?  9  gallons?  8 
gallons?  7  gallons?  6~gallons?  5  gallons?  4  gallons? 

16.  If  you  buy  5  marbles  for  a  cent,  how  many  can 
you  buy  for  12  cents?  for   11  cents?   10  cents?  9  cents? 

8  cents  ?  7  cents  ?  6  cents  ?  5  cents  ?  4  cents  ? 

17.  In   New  England  a  dollar  contains  6  shillings: 
how  many  shillings  do  12  dollars  contain  ?   1 1  dolls.  ?   10 
dolls.  ?  9  dolls.  ?  8  dolls.  ?  7  dolls.  ?  6  dolls.  ?  5  dolls.  ? 

18.  If  7   pounds  of  sugar  cost  a  dollar,  how   many 
pounds  can  you  buy  for  12  dollars?  for  11   dollars?   10 
dolls.  ?  9  dolls.?  8  dolls.?  7  dolls.  ?  6  dolls.  ?  5  dolls.? 

19.  In  New  York  a  dollar  contains  8  shillings:  how 
many  shillings  do  12  dollars  contain  ?  1 1  dolls.  ?  10  dolls.  ? 

9  dolls.  ?  8  dolls.  ?  7  dolls.  ?  6  dolls.  ?  5  dolls.  ? 

20.  At  9  cents  a  quart,  what  will   1 2  quarts  of  black- 
berries cost?   11  quarts?   10  quarts?  9  quarts?  8  quarts? 

7  quarts  ?   6  quarts  ?  5  quarts  ? 

21.  What  will  be  the  cost  of  12  yards  of  silk  at  10 
shillings  per  yard  ?  of  1 1  yards  ?    10   yards  ?  9  yards  1 

8  yards  ?  7  yards  ?  6  yards  ?  5  yards  ? 

22.  What  cost  8  cords  of  wood,  at  5  dollars  per  cord? 

23.  If  7  yards  of  cloth  make  a  cloak,  how  many  yards 
will  it  take  to  make  8  cloaks  ? 

24.  What  cost  9  pounds  of  ginger,  at  8  cents  a  pound  5 

25.  At  12  dollars  apiece,  what  will  10  cows  cost? 


.  41.]  .        MULTIPLICATION.  57 

26.  What  cost  10  barrels  of  cider,  at  9  shillings  a  bar- 
rel? 

27.  What  will  11  pair  of  shoes  come  to,  at  10  shillings 
\  pair  ? 

28.  If  8  men  can  do  a  job  of  work  in  9  days,  how  long 
will  it  take  1  man  to  do  it  ? 

29.  If  a  barrel  of  beer  will  last  7  persons  8  weeks,  how 
long  will  ii  last  1  person  ? 

30.  How  muc"h  will  3  cows  cost,  at  14  dollars  apiece? 

Analysis. — 14  is  composed  of  1  ten  and  4  units,  or  10 
and  4.  Now,  3  times  10  are  30,  and  3  times  4  are  12; 
but  12  added  to  30  make  42.  Hence,  3  times  14  dollars 
are  42  dollars.  Ans.  42  dollars. 

31.  What  cost  5  tons  of  hay,  at  13  dollars  per  ton? 

32.  What  cost  4  hogsheads  of  molasses,  at  15  dollars 
per  hogshead  ? 

33.  How  much  can  a  man  earn  in  6  months,  at  15  dol- 
lars per  month  ? 

34.  A  butcher  bought  6  sheep,  at  17  shillings  apiece, 
how  many  shillings  did  they  come  to  ? 

35.  If  a  scholar  performs  18  examples  in  1  day,  how 
many  can  he  perform  in  5  days  ? 

36.  In  1  pound  there  are  16  ounces:  how  many  ounces 
are  there  in  8  pounds  ? 

37.  How  far  will  a  man  walk  in  5  days,  if  he  walks  20 
miles  per  day  ? 

38.  If  19  men  can  build  a  house  in  4  days,  how  long 
would  it  take  one  man  to  do  it  ? 

39.  If  a  shoemaker  packs  16  pair  of  boots  in  1  box, 
how  many  pair  can  he  pack  in  7  boxes  ? 

40.  If  1  acre  of  land  produces  23  bushels  of  wheat, 
how  many  bushels  will  4  acres  produce  ? 

Analysis. — 23  is  composed  of  2  tens-  and  3  units,  or  20 
and  3.  Now  4  times  20  are  80 ;  4  times  3  are  12 ;  and 
12  and  80  are  92.  Ans.  92  bushels. 

41.  A  merchant  bought  4  pieces  of  silk,  each  piece 
having  24  yards :  how  many  yards  did  they  all  contain  ? 

42.  What  will  6  sleighs  cost,  at  25  dollars  apiece  ? 

43.  What  cost  4  reading  books,  at  42  cents  apiece  ? 


68  MULTIPLICATION.  [SECT.  IV. 

44.  In  1  guinea  there  are  21  shillings:  how  many  shil> 
lings  are  there  in  5  guineas  ? 

45.  In  1  hogshead  there  are  63  gallons:  how  many 
gallons  are  there  in  4  hogsheads  ? 

46.  What  cost  32  pounds  of  sugar,  at  8  cents   pei 
pound  ? 

47.  What  cost  85  reams  of  paper,  at  3   dollars  per 
ream  ? 

48.  What  cost  90  hats,  at  4  dollars  apiece  ? 

49.  In  1  week  there  are  7  days  :  how  many  days  are 
there  in  70  weeks? 

50.  In  1  hour  there  are  60  minutes  :  how  many  min- 
utes are  there  in  9  hours?    • 


Let  us  now  attend  to  the  nature  of  the  preceding 
operations  in  this  section.  Take,  for  instance,  the  first 
example.  Since  1  pencil  costs  4  cents,  3  pencils  will 
cost  3  times  4  cents.  Now  3  times  4  cents  is  the  same 
as  4  cents  added  to  itself  3  times  ;  and  4  cents  +  4  cents 
•j-  4  cents  are  12  cents. 

Again,  in  the  second  example  :  since  1  yard  of  cloth 
costs  6  dollars,  4  yards  will  cost  4  times  6  dollars  :  and  4 
times  6  dollars  is  the  same  as  6  dollars  added  to  itself  4 
times  ;  and  6  dollars  -j-  6  dollars  +  6  dollars  -f-  6  dollars 
are  24  dollars. 

43»  This  repeated  addition  of  a  number  or  quantity  to 
itself,  is  called  MULTIPLICATION. 

The  number  to  be  repeated  or  multiplied,  is  called  the 
multiplicand. 

The  number  by  which  we  multiply,  or  which  shows 
how  many  times  the  multiplicand  is  to  be  repeated,  is 
called  the  multiplier. 

The  number  produced,  or  the  answer  to  the  question,  is 
called  the  product. 

QUEST.  —  43.  What  is  multiplication  ?  What  is  the  number  to  be 
repeated  called  ?  What  the  number  by  which  we  multiply  ?  What 
does  the  multiplier  show  ?  What  is  the  number  produced  called  ? 
When  we  say,  6  times  12  are  72,  which  is  the  multiplicand  ?  Which 
the  multiplier  ?  Which  the  product  ? 


ARTS.  42-45.]         MULTIPLICATION.  59 

Thus,  when  we  say,  6  times  12  are  72,  12  is  the  mul- 
tiplicand, 6  the  multiplier,  and  72  the  product. 

OBS.  When  the  multiplicand  denotes  things  of  one  denomination 
only,  the  operation  is  called  Simple  Multiplication. 

44.  The  multiplier  and  multiplicand  together  are  of- 
ten called  factors}  because  they  make  or  produce  the  pro- 
duct 

Note. — The  term  factor,  is  derived  from  a  Latin  word  which  sig- 
nifies an  agent,  a  doer,  or  producer. 

45.  Multiplying  by  1,  is  taking  the  multiplicand  once: 
thus,  4  multiplied  by  1=4. 

Multiplying  by  2,  is  taking  the  multiplicand  twice :  thus, 
2  times  4,  or  4+4=8. 

Multiplying  by  3,  is  taking  the  multiplicand  three  times : 
thus  3  times  4,  or  4+4+4=12,  &c.  Hence, 

Multiplying  by  any  whole  number,  is  taking  the  multi- 
plicand as  many  times,  as  there  are  units  in  the  multiplier. 

Note. — The  application  of  this  principle  to  fractional  multipliers, 
will  be  illustrated  under  fractions, 

OBS.  1.  From  the  definition  of  multiplication,  it  is  manifest  that  the 
product  is  of  the  same  kind  or  denomination  as  the  multiplicand ; 
for,  repeating  a  number  or  quantity  does  not  alter  its  nature.  Thus, 
if  the  multiplicand  is  an  abstract  number ;  that  is,  a  number  which 
does  not  express  money,  yards,  pounds,  bushels,  or  have  reference  to 
any  particular  object,  the  product  will  be  an  abstract  number;  if  the 
multiplicand  is  money,  the  product  will  be  money ;  '^weight,  the  pro- 
duct will  be  weight;  if  measure,  measure,  &c. 

2.  Every  multiplier  is  to  be  considered  an  abstract  number.  In 
lamiliar  language  it  is  sometimes  said,  that  the  price  multiplied  by  the 
weight  will  give  the  value  of  an  article ;  and  it  is  often  asked  "how 
much  25  cents  multiplied  by  25  cents  will  produce.  But  these  are 
abbreviated  expressions,  and  are  liable  to  convey  an  erroneous  idea, 
or  rather  no  idea  at  all.  If  taken  literally,  they  are  absurd ;  for  mul- 
tiplication is  repeating  a  number  or  quantity  a  certain  number  of  times. 
Now  to  say  that  the  price  is  repeated  as  many  times  as  the  given 

QUEST. — When  we  sav,  6  times  9  are  54,  what  is  the  6  called  ?  The 
9  ?  The  54  ?  44.  What  are  the  multiplicand  and  multiplier  togethe. 
called  ?  Why  ?  Note.  What  does  the  term  factor  signify  ?  45.  Whjr, 
is  it  to  multiply  by  1  ?  By  2 1  By  3  ?  What  is  it  to  multiply  by  -^y 
whole  number?  Of  what  denomination  is  the  product?  Ho^<  does 
this  appear  ?  What  must  every  multiplier  be  considered  1  Can  you 
multiply  by  a  given  weight,  a  measure,  or  a  sum  «rf  moncv  * 


60  MULTIPLICATION.  [SECT.  IV. 

quantity  is  hazvy,  or  that  25  cents  are  repeated  25  cents  '\rnes,  is  non- 
sense. But  we  can  multiply  the  price  of  1  pound  by  a  number  equa 
to  the  number  of  pounds  in  the  -weight  of  the  given  article,  and  the 
product  will  be  the  value  of  the  article.  We  can  also  multiply  5 
cents  by  the  number  5 ;  that  is,  repeat  5  cents  5  times,  and  the  pro- 
duct is  25  cents.  Construed  in  this  manner,  the  multiplier  becomes 
an  abstract  number,  and  the  expressions  have  a  consistent  meaning. 

46 .  Multiplication  is  often  denoted  by  two  oblique  lines 
crossing  each  other  (x),  called  the  sign  of  multiplication. 
It  shows  that  the  numbers  between  which  it  is  placed, 
are  to  be  multiplied  together.     Thus  the  expression  9x6, 
signifies  that  9  and  6  are  to  be  multiplied  together,  and  is 
read,  "  9  multiplied  by  6,"  or  simply,  «  9  into  6." 

OBS.  The  product  will  be  the  same,  whether  we  multiply  9  by  6, 
or  6  by  9 ;  for,  by  the  table,  6  times  9  are  54,  also  9  times  6  are  51 
So  6X4=4X6;  5X3-3X5;  8X7-7X8,  &c. 

To  illustrate  this  point ;  suppose  there  is  a  certain  orchard  which 
contains  4  rows  of  trees,  and  each  row  has  6  trees. 
Let  the  number  of  rows  be  represented  by  the  num-     ^  % 
ber  of  horizontal  rows  of  stars  in  the  margin,  and 
the  number  of  trees  in  each  row  by  the  number  of 
stars  in  a  row.     Now  it  is  evident,  that  the  whole 
number  of  trees  in  the  orchard  is  equal  either  to  the  number  of  stars 
in  a  horizontal  row  repeated  four  times,  or  to  the  number  of  stars  in 
a  perpendicular  row  repeated  six  times;  that  is,  equal  to  6X4,  or 
4X6.     Hence, 

47 .  The.  product  of  any  two  numbers  will  be  the  same, 
whichever  factor  is  taken  for  the  multiplier. 


EXERCISES   FOR   THE   SLATE. 

Ex.  1.  What  will  3  house-lots  cost,  at  231  dollars  each  ? 

Suggestion. — If  1  house-lot  costs  23 1  dollars,  3  lots  will 
cost  3  times  231  dollars;  that  is,  three  lots  will  cosl 
231+231+231,  or  693  dollars. 


QUEST. — 46.  How  is  multiplication  sometimes  denoted  ?  What  doe* 
the  sign  of  multiplication  show  ?  How  is  the  expression  9X6,  read  ? 
How  6X7=42?  47.  Does  it  make  any  difference  in  the  prodtwt, 
Wliteh  factor  Is  made  lhe  multiplier !  How  illustrate  this ! 


ARTS.  46,  47.]          MULTIPLICATION.  61 

Having  written  the  numbers  Operation. 

upon  the  slate,  as  in  the  margin,  23 1  Multiplicand, 

we  proceed  thus :  3  times  1  unit  3  Multiplier, 

are  3  units.     Set  the  3  in  units'    ^  n    ..no  „     ,    J 
place  under  the  multiplier.     3    Dolk  693  Product- 
times  3  tens  are  9  tens ;  set  the  9  in  tens'  place.     3  times 
2  hundreds  are  6  hundreds ;  set  the  6  in  hundreds'  place. 
The  product  is  693  dollars. 

2.  What  will  4  horses  cost,  at  120  dollars  apiece? 

Suggestion. — Write  the  less  number  under  the  greater, 
and  proceed  as  before.  Ans.  480  dolls. 

3.  What  is  the  product  of  312  multiplied  by  3  ? 

Ans.  936. 

4.  What  is  the  product  of  121  multiplied  by  4  ? 

Ans.  484. 

5.  In  1  mile  there  are  320  rods :  how  many  rods  are 
there  in  3  miles? 

6.  If  a  man  travels  110  miles  in  1  day,  how  far  can  he 
travel  in  8  days  ? 

7.  8.  9.  10. 

Multiplicand,  3032         22120  101101         3012302 

Multiplier,  3453 


11.    What  will  6  stage-coaches  cost,  at  783   dollars 
n  piece  ? 

Proceeding  as  before,   6  times  3     Operation, 
units  are   18  units,  or  simply  say,  6  733 

times  3  are  18.     Now  1 8  requires  two  6 

figures  to  express  it;   hence,  we  set     A      -777^  j  i, 
the  8  under  the  figure  multiplied,  and    Ans'  4698  dolls' 
reserving  the  1,  carry  it  to  the  next  product,  as  in  addi- 
tion. (Art.  25.)     6  times  8  are  48,  and  1  (to  carry)  makes 
49.     Set  the  9  under  the  figure  multiplied,  and  carry  the 
4  to  the  next  product,  as  before.     6  times  7  are  42,  and 
4  (to  carry)  make  46.     Since  there  are  no  more  figures 
to  be  multiplied,  set  down  the  46  in  full.     The  product  ia 
4698  dollars.     Hence, 


62  MULTIPLICATION.  [SECT.    IV 

4Sf,  When  the  multiplier  contains  but  one  figure. 

Write  the  multiplier  under  the  multiplicand ;  then,  be* 
ginning  at  the  right  hand,  multiply  each  figure  of  the  mul 
tiplicand  by  the  multiplier  separately.  If  the  product  of 
any  figure  of  the  multiplicand  into  the  multiplier  docs  not 
exceed  9,  set  it  in  its  proper  place  wider  the  figure  multiplied; 
but  if  it  does  exceed  9,  write  the  U7iits>  figure  under  the 
figure  multiplied,  and  carry  tJie  lens  to  the  next  product  on 
the  left,  as  in  addition.  (Art.  25.) 

5  O.  The  principle-  of  carrying  the  tens  in  multiplica- 
tion is  the  same  as  in  addition,  and  may  be  illustrated  in 
a  similar  manner.  (Art.  26.) 

'  Take,  for  instance,  the  last  exa*  .pie,  and  set  the  pro- 
duct of  each  figure  in  a  separate  line. 

Thus.  783  '  Or,  separate  the  multiplicand  into 

6  the  orders  of  which  it  is  composed  • 

~T8  units,  ^us,  783-700+80+3 

48*  tens,  Now  700x6=4200  hund. 

42**  hunds.  80x6=-  480  tens. 

4698  Prod.  3x6= 18  units. 

Adding  these  results,  we  have'  4698  Product. 

In  this  analytic  solution  it  will  be  seen  that  the  tens'" 
figure  in  each  product  which  exceeds  9,  is  added  to  the 
next  product  on  the  left,  the  same  as  in  the  common  meth- 
od of  solving  this  and  similar  examples.  The  only  dif- 
ference between  the  two  operations  is,  that  in  one  case  we 
add  the  tens  as  we  proceed  in  the  multiplication  ;  in  the 
other  we  reserve  them  till  each  figure  is  multiplied,  and 
then  add  them  to  the  same  orders  as  before  :  consequently, 
the  result  must  be  the  same  in  both.  (Art.  27.) 


QUEST.— 49.  How  do  you  write  the  numbers  for  multiplication 
Where  begin  to  multiply  ?    When  the  product  of  a  figure  in  the  ^nul 
tiplicand  does  not  exceed  9,  where  is  it  written  ?     When  it  exceeds  9 
what  is  to  be  done  with  it  ?    50.  How  illustrate  the  principle  of  car 
tying  in  multiplioaticm  \ 


ARTS.  49-51.]          MULTIPLICATION.  63 

51*  From  this  and  the  preceding-  illustrations,  the 
earner  will  perceive,  that  units  multiplied  by  units  pro- 
luce  units ;  tens  into  units  produce  tens ;  hundreds  into 
units  produce  hundreds,  &c.  Hence, 

When  the  multiplier  is  units,  the  product  will  always  be 
of  the  same  order  as  the  figure  multiplied. 

12.  What  cost  83  pounds  of  opium,  at  8  dollars  per 
pound  ? 

13.  At  9  shillings  per  day,  how  much  can  a  man  earn 
in  213  days? 

14.  If  1  sofa  costs  78  dollars,  Mbw  much  will  8  sofa* 
cost? 

1 5.  What  cost  879  barrels  of  flour,  at  7  dollars  a  barrel? 

16.  At  8  shillings  apiece,  what  will  a  drove  of  650 
lambs  come  to  ? 

17.  18.  19.  20. 

Multiply     8006  76030  10906  4608790 

By  5  8  7  9 


21.  What  will  26  horses  cost,  at  113  dollars  apiece? 

Suggestion. — Reasoning  as  before,  if  1  horse  costs  113 
dollars,  26  horses  will  cost  26  times  as  much. 

Since  it  is  not  conven-         Operation. 
lent  to  multiply  by  26  at  113  Multiplicand, 

once,   we   first    multiply  26  Multiplier, 

oy  the  6  uftits,  then  by  -g78  cost  of  6  horses, 

the  2  tens  and   add  the  ggS*  cost  of  20     " 

two    results     together. — 

Thus  6  times  3  are  18;  Ans-  2938  cost  of  26  " 
set  doAvn  the  8  and  carry  the  1,  as  above.  6  times  1  are 
6,  and  1  to  carry  makes  7.  .6  times  1  are  6.  Next,  mul- 
tiply by  the  2  tens  thus :  20  "times  3  unitb  are  60  units  or 
6  tens  ;  or  we  may  simply  say,  2  times  3  are  6.  Now 
the  6  must  denote  tens ;  for  units  into  tens,  or  what  is 


QUEST. — 51.  What  do  units  multiplied  into  units  produce  ?  Tana 
into  units  ?  Of  what  order  is  the  product  universally,  when  the  multl- 
olier  is  units  ? 


64  MULTIPLICATION.  [SECT.  IV 

the  same  thing,  (Art.  47,)  tens  into  units,  produces  tens  . 
consequently  the  6  must  be  written  in  tens'  place  in  the 
product;  that  is,  under  the  figure  2  by  which  we  are 
multiplying.  20  times  1  ten  are  20  tens  or  200 ;  or  sim- 
ply say,  2  times  1  are  2 :  and  since  the  2  denotes  hun- 
dreds, as  we  have  just  seen,  set  it  on  the  left  of  the  6  in 
hundreds'  place.  20  times  1  hundred  are  20  hundred  01 
2000 ;  or  simply  say,  2  times  1  are  2 :  and  since  the  2 
denotes  thousands,  set  it  in  the  thousands'  place  on  the  left 
of  the  last  figure  in  the  product.  Finally,  adding  these 
two  results  together  as  they  stand,  units  to  units,  tens  to 
tens,  &c.,  we  have  2938  dollars,  which  is  the  whole  pro- 
duct required. 

Note. — The  several  products  of  the  multiplicand  into  the  separata 
figures  of  the  multiplier,  are  called  partial  products.  Hence, 

52.  When   the   multiplier   contains   more    than   one 
figure. 

Multiply  each  figure  of  the,  multiplicand  by  each  figure 
of  the  multiplier  separately,  and  write,  each  partial  product 
in  a  separate  line,  placing  the  first  figure  of  each  line  directly 
under  that  by  which  you  multiply;  finally,  add  the  several 
partial  products  together,  and  the  sum  will  be  the  true  pro- 
duct or  answer  required. 

53.  PROOF. — Multiply  the  multiplier  by  the  multipli- 
cand, and  if  the  product  thus  obtained  is  the  same  as  the  other 
product,  the  work  is  supposed  to  be  right. 

OBS  1.  This  method  of  proof  depends  upon  the  principle,  that  the 
product  of  any  two  numbers  is  the  same,  whichever  is*  taken  for  the 
multiplier.  (Art.  47.) 

2.  When  the  multiplier  is  small,  we  may  add  the  multiplicand  to 
itself  as  many  times  as  there  are  units  in  the  multiplier,  and  if  the 
sum  is  equal  to  the  product,  the  work  is  right.     Thus  78X3=234. 
Proof. — 78-|-78-(-78=:234,  which  is  the  same  as  the  product. 

3.  Multiplication  may  also  be  proved  by  division,  and  by  casting 
yitt  the  nines ;  but  neither  of  these  methods  can  be  explained  her* 

QUEST. — Note.  What  is  meant  by  partial  products  ?  52.  How  da 
you  proceed  when  the  multiplier  contains  more  than  one  figure  ?  How 
should  the  partial  products  be  written  ?  Where  write  the  first  figure 
of  each  line  ?  What  do  you  finally  do  with  the  partial  products  ? 


53.  How  is  multiplication  proved  ?     Obs.  On  what  principle  c 
method  of  proof  depend  ?    When  the  multiplier  is  small,  hew 


)le  does  tliii 
may  we 
prove  it  I 


ARTS.  52,  53.]          MULTIPLICATION.  65 

without  anticipating  principles  belonging  to  division,  with  which  the 
learner  is  supposed  as  yet  to  be  unacquainted. 

22.  What  will  45  cows  cost,  at  27  dollars  a  head? 

Operation.  Proof. 

45  Multiplicand,  27 

27  Multiplier,  45 

315  135 

90  108 


1215  Product.  1215  Product. 

23.  What  cost  63  hats,  at  36  shillings  apiece  ? 

24.  How  much  corn  can  a  man  raise  on  87  acres,  at 
45  bushels  per  acre  ? 

25.  How  many  pounds  of  sugar  will  75  boxes  contain, 
if  each  box  holds  256  pounds  ? 

26.  What  cost  278  hogsheads  of  molasses,  at  23  dol 
tars  per  hogshead  ? 

27.  What  is  the  product  of  347  multiplied  by  256? 

Operation. 

Suggestion. — Proceed  in  the  same  347 

manner  as  when  the  multiplier  con-  256 

tains  but  two  figures,  remembering  to  2082 

place  the   right  hand  figure   of  each  1735 

partial  product  directly  under  the  fig-  594 

ure  by  which  you  multiply.  ••           •     . 

28.  What  is  the  product  of  569  into  308  ? 

After  multiplying  by  the  8  units,         Operation. 
»ve  must  next -multiply  by  the  3  hun-  569 

dreds,  since  there  are  no  tens  in  the  308 

multiplier,  and  place  the  first  figure 
of  this  partial  product  directly  under 
the  figure  3  by  which  we  are  multi- 
plying. 

29.  What  is  the  product  of  67025  into  4005  ? 

AM.  268435125. 
80.  What  is  the  product  of  841072  mto  603  2 


£6  MULTIPLICATION.  [SECT.  IV 

54.  From  the  preceding  illustrations  and  principles 
we  derive  the  following 

GENERAL  RULE  FOR  MULTIPLICATION. 

I.  Write  the  multiplier  under  the  multiplicand,  units 
under  units,  tens  under  tens,  fyc. 

II.  When  the  multiplier  contains  but  one  figure. 
Begin  with,  the  units,  and  multiply  each  figure  of  the 

multiplicand  by  the  multiplier,  setting  down  the  result  and 
carrying  as  in  addition.    (Art.  49.) 

III.  When    the   multiplier    contains  more  than   one 
figure. 

Multiply  each  figure  of  the  multiplicand  by  each  figure 
of  the  multiplier  separately,  beginning  at  the  right  hand, 
and  write  the  partial  products  in  separate  lines,  placing  the 
first  figure  of  each  line  directly  under  the  figure  by  which 
you  multiply.  (Art.  52.) 

Finally,  add  the  several  partial  product*  together,  and-  the 
sum  will  be  the  whole  product. 

OBS.  It  is  immaterial  as  to  the  result  which  of  the  factors  is  taken 
for  the  multiplier.  (Art.  47.)  But  it  is  more  convenient  and  therefore 
customary  to  place  the  larger  for  the  multiplicand  and  the  smaller  for 
the  multiplier.  Thus,  it  is  easier  to  multiply  254672381  by  7,  than 
it  is  to  multiply  7  by  254672381,  but  the  product  will  be  the  same. 

EXAMPLES   FOR    PRACTICE. 

1.  What  will  465  hats  cost,  at  6  dollars  apiece? 

2.  What  will  638  sheep  cost,  at  4  dollars  a  head? 

3.  What  will  1360  yards  of  cloth  cost,  at  7  dollars  a 
yard? 

4.  What  cost  169  bushels  of  potatoes,  at  4  shillings  per 
bushel  ? 

5.  What  cost  279  barrels  of  salt,  at  9  shillings  a  barrel  ? 

6.  At  12  dollars  a  suit,  how  much  will  it  cost  to  fur- 
nish 1161  soldiers  with  a  suit  of  clothes  apiece? 

7.  What  cost  1565  acres  of  wild  land,  at  7  dollars  pel 
acre? 


QUEST. — 54.   What  is  the  general  rule  for  multiplication?    Obs. 
Which  number  is  usually  taken  for  the  multiplicand  1 


A.E.T.  54.]  MULTIPLICATION.  67 

8.  What  will  758  baskets  of  peaches  cost,  at  5  dollars 
per  basket? 

9.  What  cost  25650  pounds  of  opium,  at  6  dollars  a 
pound  ? 

10.  How  much  can  a  man  earn  in  12  months,  at  15 
dollars  per  month  ? 

11.  What  will  23  loads  of  hay  come  to,  at  18  dollars  a 
load? 

12.  What  will  45  cows  come  to,  at  21  dollars  apiece? 

13.  What  will  56  hogsheads  of  molasses  cost,  at  32 
dollars  a  hogshead  ? 

14.  What  cost  128  firkins  of  butter,  at  13  dollars  a 
firkin  ? 

1 5.  What  cost  97  kegs  of  tobacco,  at  26  dollars  per  keg  ? 

16.  What  cost  110  barrels  of  pork,  at  19  dollars  per 
barrel  ? 

17.  How  much  will  235  sheep  come  to,  at  21  shillings 
ahead? 

18.  How  many  bushels  of  corn  will  grow  on  83  acres, 
at  the  average  rate  of  37  bushels  to  an  acre  ? 

19.  In  one  bushel  there  are  32  quarts:   how  many 
quarts  are  there  in  92  bushels  ? 

20.  What  will  a  drove  of  463  cattle  come  to,  at  48  dol- 
lars per  head  ? 

21.  How  much  will  78  thousand  of  boards  cost,  at  19 
dollars  per  thousand  ? 

22.  What  cost  243  chests  of  tea,  at  37  dollars  per 
chest? 

23.  A  man  bought  168  horses,  at  63  dollars  apiece : 
what  did  they  come  to  ? 

24.  What  cost  256  barrels  of  beef,  at  16  dollars  a 
barrel ? 

25.  If  376  men  can  build  a  fortification  in  95  days, 
how  long  would  it  take  1  man  to  build  it  ? 

26.  Allowing  365  days  to  a  year,  how  many  days  has 
a  man  lived  who  is  45  years  old? 

27.  If  a  garrison  consume  725  pounds  of  beef  in  one 
day,  how  many  pounds  will  they  consume  in  125  days? 

28.  How  many  pounds  will  the  same  garrison  con- 
eurae  in  243  days  2 


MULTIPLICATION.  [SECT.  IV. 

29.  How  far  will  a  ship  sail  in  365  days,  at  215  miles 
per  day? 

30.  What  costs  678  tons  of  Railroad  iron,  at  115  dol- 
lars per  ton  ? 

CONTRACTIONS  IN  MULTIPLICATION. 

55.  The  general  rule  is  adequate  to  the  solution  of 
all  examples  that  occur  in  multiplication.  In  many  in- 
stances, however,  by  the  exercise  of  judgment  in  apply- 
ing  the  preceding  principles,  the  operation  may  be  very 
much  abridged. 

CASE  I. —  When  t/ie  multiplier  is  a  composite  number. 
Ex.  1.  What  will  14  hats  cost,  at  8  dollars  apiece? 

Analysis. — Since  14  is  twice  as  much  as  7 ;  that  is, 
14=7x2,  it  is  manifest  that  14  hats  will  cost  twice  as 
much  as  7  hats. 

Instead  of  multiplying  by 

7  14,  we  may  first  find  the 

cost   of  7    hats,   and    then 

56  cost  of  7  hats.        multiply  that  product  by  2, 

which  will  give  the  cost  of 

Dolls.  112  cost  of  14  hats.  14  hats.  In  other  words, 
we  may  first  multiply  by  the  factor  7.  and  that  product  by 
2,  the  other  factor  of  14. 

Proof. — 14x8=112,  the  same  as  before. 

2.  What  will  27  horses  cost,  at  85  dollars  apiece  ? 

Suggestion. — Find  the  factors  of  27  ;  that  is,  find  two 
numbers,  which  being  multiplied  together,  produce  27, 
and  multiply  first  by  one  of  these  factors,  and  the  product 
thus  arising  by  the  other. 

OBS.  1.  Any  number  which  may  be  produced  by  multiplying  tw» 
or  more  numbers  together,  is  called  a  composite  number,  and  the  fac- 
tors, which  being  multiplied  together,  produce  the  composite  number, 
are  sometimes  called  the  component  parts  of  the  number.  Thus,  14 
27,  32,  &c.,  are  composite  numbers,  and  the  factors  7  and  2,  9  and  3 
8  and  4,  are  their  component  parts. 


ARTS.  55-57.]  MULTIPLICATION.  69 

2.  The  process  of  finding  the  factors  of  which  a  given  number  is 
composed,  is  called  resolving  the  number  into  factors. 

56.  Some  numbers  may  be  resolved  into  wore  than 
two  factors  ;  and  also  into  different  sets  of  factors.     Thus, 
the  factors  of  24  are  3,  2,  2  and  2  ;  or  4,  3  and  2  ;  or  6, 
2  arid  2 ;  or  8  and  3  ;  or  6  and  4  ;  or  12  and  2. 

OBS.  We  have  seen  that  the  product  of  any  two  numbers  is  the 
same,  whichever  factor  is  taken  for  the  multiplier.  (Art.  47.)  In 
like  manner,  the  product  of  any  tkree  or  more  factors  is  the  same,  in 
whatever  order  they  are  multiplied.  For,  the  product  of  two  factors, 
may  be  considered  as  one  number,  and  this  may  be  taken  either  for 
the  multiplicand,  or  the  multiplier.  Again,  the  product  of  three  fac- 
tors may  be  considered  as  one  number,  and  be  taken  for  the  multipli- 
cand, or  the  multiplier,  &c.  Thus,  24=3X2X2X2=0X2X2=12 
X2=6X4=4X2X.3=8X3. 

3.  What  will  24  hogsheads  of  molasses  cost,  at  37  dol- 
lars per  hogshead?  Ans.  838  dollars. 

Suggestion. — Resolve  24  into  any  two  or  mare  factors, 
and  proceed  as  before.  Hence, 

57.  To  multiply  by  a  composite  number. 

Resolve  the  multiplier  into  two  or  more  factors;  multi- 
ply the  multiplicand  by  one  of  these  factors,  and  this  pro- 
duct by  another  factor,  and  so  on  till  you  have  multiplied 
by  all  the  factors.  The  last  product  will  be  t/ie  product 
required. 

OBS.  The  factors  into  which  a  number  may  be  resolved,  must  not 
be  confounded  with  the  parts  into  which  it  may  be  separated.  (Art. 
26.)  The  former  have  reference  to  multiplication,  the  latter  to  ad- 
lition  ;  that  is,  factors  must  be  multiplied  together,  but  parts  must  be 
idded  together  to  produce  the  given  number.  Thus,  56  may  be  re- 
solved into  two  factors,  8  and  7 ;  it  may  be  separated  into  two  parts, 
5  tens  or  50,  and  6.  Now  8Xf — 56,  and  50-j-6=56. 

4.  What  will  36  cows  cost,  at  19  dollars  a  head? 

QUEST. — Obs.  What  is  a  composite  number?  What  are  the  factors 
which  produce  it,  sometimes  called  ?  What  is  meant  by  resolving  a 
number  into  factors  ?  56.  Are  numbers  ever  composed  of  more  than 
two  factors  ?  What  are  the  factors  of  24  ?  32  ?  36  ?  40  ?  42  ?  60  ?  64  ? 
72  ?  108  ?  Obs.  When  three  or  more  factors  are  to  be  multiplied  to- 
gether, does  it  make  any  difference  in  what  order  they  are  taken  ? 
57.  When  the  multiplier'is  a  composite  number,  how  do  you  proceed  ? 
Obs.  What  is  the  difference  between  the  factors  into  which  a  number 
may  be  resolved,  and  the  parts  into  which  it  may  bo  separated  ? 


TO  MULTIPLICATION.  [SECT.  IV, 

5.  What  cost  45  acres  of  land,  at  110  dollars  per  acre? 

6.  At  36  shillings  per  week,  how  much  will  it  cost  a 
person  to  board  52  weeks  ? 

7.  If  a  man  travels  at  the  rate  of  42  miles  a  day,  how 
far  can  he  travel  in  205  days  ? 

8.  At  the  rate  of  56  bushels  per  acre,  how  much  corn 
can  be  raised  on  460  acres  of  land  1 

9.  What  cost  672  yards  of  broadcloth,  at  24  shillings 
per  yard  ? 

10.  What  cost  1265  yoke  of  oxen,  at  72  dollars  per 
yoke  ? 

CASE  II. —  When  the  multiplier  is  1  with  ciphers  annex- 
ed to  it. 

58.  It  is  a  fundamental  principle  of  notation,  that  each 
removal  of  a  figure  one  place  towards  the  left,  increases 
its  value  ten  times;  (Art.  9 ;)  consequently,   annexing  a 
cipher  to  a  number  will  increase  its  value  ten  times,  or 
multiply  it  by   10;  annexing  two  ciphers,  will  increase 
its  value  a  hundred  times,  or  multiply  it  by  100;  annex- 
ing three   ciphers  will    increase  it   a   thousand  times,  or 
multiply  it  by  1000,  &c. ;  for  each  cipher  annexed,  re- 
moves each  figure  in  the  number  one  place  towards  the 
left.      Thus,   12  with  a  cipher  annexed,  becomes   120, 
and  is  the  same  as    12x10;    12  with   two  ciphers   an- 
nexed, becomes   1200,  and  is  the  same  as  12x100;   12 
with  Hire*  ciphers  annexed,  becomes   12000,  and  is  the 
same  as  12x1000,  &c.     Hence, 

59.  To  multiply  by  10,  100.  1000,  &c. 

Annex  as  many  ciphers  to  the  multiplicand  as  there  are 
ciphers  in  the  multiplier,  and  the  number  thus  formed  will  he 
the  product  required. 

Note. — To  annex  means  to  place  after,  or  at  the  right  hand. 

1 1.  What  will  10  drums  of  figs  weigh,  at  28  pounds  a 
drum  1  Ans.  280  pounds. 

QUEST. — 58.  What  effect  does  it  have  to  remove  a  figure  one  place 
towards  the  left  hand  ?  Two  places  ?  59.  How  do  you  proceed  when 
the  multiplier  is  10,  100,  1000,  &c.  I  Note.  What  is  the  meaning  of 
?b*  term  ami** 7 


\RTS.  58-60.]  MULTIPLICATION.  71 

12.  How  many  pages  are  there  in   100  booics,  each 
book  having  352  pages'? 

13.  Multiply  476  by  1000. 

14.  Multiply  53486  by  10000. 

15.  Multiply  12046708  by  100000. 

16.  Multiply  26900785  by  1000000. 

17.  Multiply  89063457  by  10000000. 

18.  Multiply  9460305068  by  100000. 

19.  Multiply  78312065073  by  10000. 

CASE  III. — When  the  multiplier  has  ciphers  on  ike  right. 

20.  What  will  20  acres  of  land  cost,  at  32  dollars  per 
acre? 

Note. — Any  number  with  ciphers  on  its  right  hand,  is  obviously  a 
romposite  number ;  the  significant  figure  or  figures  being  one  factor, 
and  1  with  the  given  ciphers  annexed  to  it,  the  other  factor.  Thus 
20  may  be  resolved  into  the  factors  '2  and  1 0.  We  may  therefore  first 
multiply  by  2  and  then  by  10,  by  annexing  a  cipher  as  above. 

Solwtwn.— 32x2=64,  and  64x10-640  dolls.  Ans. 

•21.  If  the  expenses  of  an  army  are  2000  dollars  per 
day,  what  will  it  cost  to  support  the  same  army  365 
days? 

n        .  •  2000  may  be  resolved  into  the  factors  ^  and 

365  1?00'     Then  2  times  3(?5  are  73°  5  now  ?d- 

o  ding  three  ciphers  to  this  product,  multiplies 

—£  it  by  1000,  (Art.  59,)  and  we  have  730000 

730000     (10^^  for  the  answer.     Hence, 


6O.  When  there  are  ciphers  on  the  right  hand  of  the 
multiplier. 

Multiply  the  multiplicand  by  the  significant  figures  of  the 
multiplier,  and  to  this  product  annex  as  many  ciplters  as 
are  found  on  the  right  hand  of  the  multiplier. 

OF,S.  It  will  be  perceived  that  this  case  combines  the  principles  of 
the  two  preceding  cases;  for,  the  multiplier  is  a  composite  number 
and  one  of  its  factors  is  1  with  ciphers  annexed  to  it. 

QUEST. — GO.  When  there  are  ciphers  on  the  riaht  of  the  multiplier, 
ho'.v  do  you  proceed  !  O.V.  What  principles  a~e.  combine*!  in  this  r,ase  t 


72  MULTIPLICATION.  [SECT.  IV 

22.  How  many  days  are  there  in  36  months,  reckon 
ing  30  days  to  a  month  ? 

23.  If  1  barrel  of  flour  weighs  192  pounds,  how  much 
will  200  barrels  weigh  ? 

24.  Multiply  4376  by  2500. 

25.  Multiply  50634  by  4 1000. 

26.  Multiply  630125  by  620000. 

CASE  Ju  —  When  the  multiplicand  has  cipJiers  on  tfo. 
right. 

27.  Multiply  12000  by  31. 

Suggestion. — 12000    is    a   composite  Operation. 

number,  the  factors  of  \vhich  are  12  and  12000 

1000.     But  the  product  of  two  or  more  31 

numbers  is  the  same  in  whatever  order  j2 

they  are  multiplied ;  (Art.  47  ;)  conse-  gg 

quently  multiplying  the  factor  12  by  31, 
and  this  product  by  1000,  will  give  the        A™-  3'20UO 
same  result  as  12000x31.     Thus,  31  times  12  are  372; 
then  annexing  three  ciphers,  we  have  372000,  which  is 
the  same  as  12000x31.     Hence, 

6 1 .  When  there  are  ciphers  on  the  right  of  the  mul- 
tiplicand. 

Multiply  the  significant  figures  of  the  multiplicand  by 
the  multiplier,  and  to  the  product  annex  as  many  ciphers  as 
are  found  on  the  right  of  the  multiplicand. 

OBS.  When  the  multiplier  and  multiplicand  both  have  ciphers  on 
the  right,  multiply  the  significant  figures  together,  and  to  their  pro- 
duct annex  as  many  ciphers  as  are  found  on  the  right  of  both  factors. 

28.  Multiply  370000  by  32. 

29.  Multiply  8120000  by  46. 

30.  Multiply  56300000  by  64. 

31.  Multiply  623000000  by  89. 

32.  Multiply  54000  by  700.  Ans.  37800000. 


QUEST. — 61.  When  there  are  ciphers  on  the  right  hand  of  the  multi- 
plicand, how  proceed  ?  Obs.  How,  when  there  are  ciphers  en  th« 
right  both  of  the  multiplier  and  multiplicand  ? 


ARTS.  61-63.]  DIVISION.  73 

33.  Multiply  4300  by  600.  Ans.  2580000. 

34.  Multiply  563800  by  7200. 

35.  Multiply  1230000  by  12000. 

36.  Multiply  310200  by  20000. 

37.  Multiply  2065000  by  810000. 

38.  Multiply  2109090  by  510000. 


SECTION    V. 
DIVISION. 

MENTAL    EXERCISES. 

ART.  63.  Ex.  1.  How  many  oranges,  at  3  cents 
apiece,  can  you  buy  for  12  cents'? 

Suggestion. — If  3  cents  buy  one  orange,  12  cents  will 
buy  as  many  oranges  as  there  are  3  cents  in  12  cents; 
that  is,  as  many  as  3  is  contained  times  in  12.  Now  3  is 
contained  in  12,  4  times.  .Arcs.  4  oranges. 

2.  How  many  lemons,  at  4  cents  apiece,  can  you  bu> 
for  20  cents  ? 

Suggestion. — To  find  how  many  times  4  cents  are 
contained  in  20  cents,  think  how  many  times  4  make  20, 
or  what  number  multiplied  by  4,  produces  20. 

3.  At  3  dollars  per  yard,  how  many  yards  of  cloth  can 
be  bought  for  15  dollars? 

4.  How  many  hats,  at  5  dollars  apiece,  can  you  buy 
for  30  dollars  ? 

5.  How  many  barrels  of  flour  will  36  bushels  of  wheat 
make,  allowing  4  bushels  to  one  barrel  ? 

6.  If  you  pay  6  cents  a  mile  for  riding  in  a  stage,  h  j  kv 
far  can  you  ride  for  48  cents? 

7.  If  a  pound  of  sugar  cost  7  cents,  how  many  pounds 
can  you  buy  for  56  cents. 


74  DIVISION.  [SECT.  V 

8.  How  many  slates,  at  8  cents    apiece,  can  you  buy 
for  40  cents  ? 

9.  Four  quarts  make  one  gallon :  how  many  gallons 
are  there  in  48  quarts  ? 

10.  At  7  dollars  a  ton,  how  many  tons  of  coal  can  b« 
bought  for  63  dollars  ? 


DIVISION  TABLE. 


2  in 

3in 

4  in 

5  in 

6  in 

7  in 

8  in 

9  in    | 

2,  once 

3,once 

4,  once 

5,  once 

6,once 

7,  once 

8,  once 

9,  once) 

4,      2 

6,      2 

8,      2 

10,      2 

12,      2 

14,      2 

]6,      2 

18,      2 

6,      3 

9,      3 

12,      3 

15,      3 

18,      3 

21,      3 

24,      3 

27,      3. 

8,      4 

12,      4 

16,      4 

20,      4 

24,      4 

28,      4 

32,      4 

36,      4 

10,      5 

15,      5 

20,      5 

25,      5 

30,      5 

35,      5 

40,      5 

45,      5 

12,      6 

18,      6 

24,      6 

30,      6 

36,      6 

42,      6 

48,      6 

54,      6 

14,      7 

21,      7 

28,      7 

35,      7 

42,      7 

49,      7 

56,      7 

63,      7) 

16,      8 

24,      8 

32,      8 

40,      8 

48,      8 

56,      8 

64,      8 

72,    s; 

18,      9 

27,      9 

36,      9 

45,      9 

54,      9 

63,      9 

72,      9 

81,      9: 

11.  How  many  pair  of  boots,  at  2  dollars  a  pair,  can 
be  bought  for  24  dollars?   for  22?  20?   18?  16?' 14? 
12?   10? 

12.  How  many  barrels  of  cider,  at  3  dollars  a  barrel, 
can  you  buy  for  36  dollars?  for  30?  27?  24?  21?   18? 
15?   12? 

13.  How  many  quarts  of  milk,  at  4  cents  a  quart,  can 
you  buy  for  48  cents?  for  44?  40?  36?  32?  28?  24? 
20?   16? 

14.  At  5  cents  an  ounce,  how  many  ounces  of  wafers 
can  you  buy  for  60  cents  ?  for  55  ?  50  ?  45  ?  40  ?  35  ? 
30?  25? 

15.  At  6  shillings  a  pair,  how  many  pair  of  gloves  can 
be  bought  for  60  shillings?  for  54?  48?  42?  36?  30? 
24?   18? 

16.  How  many  pounds  of  butter,  at  7  cents  a  pound, 
can  be  purchased  for  63  cents?  56?  49?  42?  35?  28? 
21?   14? 

17.  How  many  cloaks  will  72  yards  of  cloth  make, 
allowing  8  yards  to  a  cloak  ?  how  many  64  ?  56  ?  48  ? 
40?  32?  24? 

18.  How   many   cows,   at  9   dollars  apiece,   can   be 


ART.  63.]  DIVISION.  75 

bought  for  81  dollars?  for  72?  63?  54?  45?  36?  27? 
18?  9? 

19.  How  many  times  is  4  contained  in  36  ?  48  ?  40  ? 

20.  How  many  times  is  8  contained  in  40?  56?  48? 
64?  72? 

21.  In  25,  how  many-'times  4,  and  how  many  over? 

Ans.  6  times  and  1  over. 

22.  In  34,  how  many  times  5,  and  how  many  over  ? 
In  43?  45?  37?  28?  39? 

23.  In  23,  how  many  times  3,  and  how  many  over  ? 
How  many  times  4  ?  2  ?   10  ?  6  ? 

24.  In  24,  how  many  times  7,  and  how  many  over  ? 
6?  5?  9?  12?  2? 

25.  In  36,  how  many  times  6?  7?  3?  8?   12?  5?  9? 

26.  In  32,  how  many  times  6?  4?  3?  16? 

27.  How  many  hats,  at  6  dollars  apiece,  can  be  bought 
for  60  dollars? 

28.  How  many  tons  of  hay,  at  9  dollars  per  ton,  can 
you  buy  for  81  dollars. 

29.  If  you  travel  7  miles  an  hour,  how  long  will  it 
fake  to  travel  70  miles  ? 

30.  If  you  pay  10  cents  apiece  for  slates,  how  many 
can  you  buy  for  95  cents,  and  how  many  cents  over  ? 

31.  George  bought  12  oranges,  Avhich  he  wishes  to  di- 
vide equally  between  his  2  brothers :  how  many  can  he 
give  to  each? 

Suggestion. — Since  there  are  12  oranges  to  be  divided 
equally  between  2  boys,  each  boy  must  receive  1  orange 
as  often  as  2  oranges  are  contained  in  12  oranges;  that 
is.  each  must  receive  as  many  oranges  as  2  is  contained 
times  in  12.  But  2  is  contained  in  12,  6  times;  for  6 
times  2  make  12. 

Ans.  6  oranges. 

32.  Henry  has  15  apples,  which  he  wishes  to  divide 
equally  among  3  01  his  companions :  how  many  can  he 
give  to  each? 

33.  A  gentleman  sent  20  peaches  to  be  divided  equally 
among  4  boys :  how  many  did  each  boy  receive  ? 

34.  A  dairy-woman  having  30  pounds  of  butter,  wish- 


76  DIVISION.  [SECT.  V. 

es  to  pack  it  in  5  boxes,  so  that  each  box  shall  have  an 
equal  number  of  pounds :  how  many  pounds  must  she 
put  in  each  box  ? 

35.  I  have  21   acres  of  land,  which  I  wish  to  fence 
into  7  equal  lots :  how  many  acres  must  I  put  into  each 
lot? 

36.  A  boy  having  28  marbles,  wished  to  divide  them 
into  4  equal  piles :  how  many  must  he  put  in  a  pile  ? 

37.  I  have  40  peach-trees,  which  I  wish  to  set  out  in  5 
3qual  rows  :  how  many  must  I  sgt  in  a  row  ? 

38.  There  were  45  scholars  in  a  certain  school,  and 
the  teacher  divided  them  into  5  equal  classes :  how  many 
did  he  put  in  a  class  ? 

39.  If  50  dollars  were  divided  equally  among-  10  men, 
how  many  dollars  would  each  man  receive  ? 

40.  A  company  of  8  boys  buying  a  boat  for  32  dollars, 
agreed  to  share  the  expense  equally:  how  much  must 
each  one  pay? 

41.  In  a  certain  orchard  there  are  54  apple-trees,  and 
5  trees  in  each  row :  how  many  rows  are  there  in  the 
orchard  ? 

42.  If  63  quills  are  divided  equally  among  7  pupils, 
how  many  will  each  receive  ? 

43.  If  you  divide  36  into  4  equal  parts,  how  many  will 
there  be  in  a  part  ? 

44.  If  you  divide  56  into  8  equal  parts,  how  many  will 
each  part  contain  ? 

45.  If  you  divide  48  into  6  equal  parts,  how  many  will 
each  part  contain  ? 

46.  A  gentleman  distributed  40  dollars  equally  among 
8  beggars  :  how  many  dollars  did  he  give  to  each  ? 

47.  A  company  of  6  boys  found  a  pocket-book,  and  on 
returning  it  to  its  owner,  he  handed  them  60  dollars  to  be 
shared  equally  among  them  :  what  was  each  one's  share  ? 

48.  A  merchant  received  72  dollars  for  6  coats  of  equal 
value :  how  much  was  that  apiece  1 

49.  A  man  paid  81  cents  for  the  use  of  a  horse  and 
ouggy  to  ride  9  miles :  how  much  was  that  a  mile  ? 

50.  If  you  divide  90  dollars  into  10  equal  paits,  how 
many  dollars  will  there  be  in  each  part  ? 


.  64-o7.]  DIVISION.  77 

OBS.  The  object  in  each  of  the  last  twenty  questions,  is  to  divide  a 
given  number  into  several  equal  parts,  and  ascertain  the  value  of  these 
parts ;  but  the  method  of  solving  them  is  precisely  the  same  as  that 
of  the  preceding  ones. 

64.  The  process  by  which  the  foregoing  examples 
are  solved,  is  called  DIVISION. 

It  consists  in  finding  how  many  times  one  given  number 
is  contained  in  another. 

The  number  to  be  divided,  is  called  the  dividend. 

The  number  by  which  we  divide,  is  called  the  divisor. 

The  number  obtained  by  division,  or  the  answer  to  the 
question,  is  called  the  quotient.  It  shows  how  many  times 
the  dividend  contains  the  divisor.  Hence,  it  may  be  said 

6  5  •  Division  is  finding  a  quotient,  which  multiplied  in- 
to the  divisor,  will  produce  the  dividend. 

Note. — The  term  quotient  is  derived  from  the  Latin  word  quatiet, 
which  signifies  how  often,  or  how  many  times. 

66.  The  number  which  is  sometimes  left  after  divis- 
ion, is  called  the  remainder.     Thus,  in  the  twenty-first  ex- 
ample, when  we  say  4  is  contained  in  25,  6  times  and  1 
over,  4  is  the  divisor,  25  the  dividend,  6  the  quotient,  and 
1  the  remainder. 

OBS.  1.  The  remainder  is  always  less  than  the  divisor;  for  if  it 
vere  equal  to,  or  greater  than  the  divisor,  the  divisor  could  be  con- 
fined once  more  in  the  dividend. 

2.  The  remainder  is  also  of  the  same  denomination  as  the  divi- 
dend ;  for  it  is  a  part  of  it. 

67.  Division  is  denoted  in  two  ways : 


QUEST. — 64.  In  what  does  division  consist?  What  is  the  number  to 
be  divided,  called  ?  The  number  by  which  we  divide  ?  What  is  the 
number  obtained,  called  ?  What  does  the  quotient  show  ?  65.  What 
then  may  division  be  said  to  be?  66.  What  is  the  number  called 
which  is  sometimes  left  after  division  ?  When  we  say  4  is  in  25,  6 
times  and  1  over,  what  is  the  4  called  ?  The  25  ?  The  6  ?  The  1  ? 
When  we  say  6  is  in  45,  7  times  and  3  over,  which  is  the  divisor? 
The  dividend ?  The  quotient?  The  remainder.  Obs.  Is  the  remain- 
der greater  or  less  than  the  divisor  ?  Why  ?  Of  what  denomination 
is  it  1  Why  ?  67.  How  many  ways  is  division  denoted  ? 


78  DIVISION.  [SECT.  V. 

First,  by  a  horizontal  line  between  two  dots  (-*-),  called 
the  sign  of  division,  which  shows  that  the  number  be- 
fore it,  is  to  be  divided  by  the  number  after  it.  Thus 
the  expression  24-J-6,  signifies  that  24  is  to  be  divided 
by  6. 

Second,  division  is  often  expressed  by  placing  the  di 
visor  under  the  dividend  with  a  short  line*  between  them 
Thus  the  expression  3T5 ,  shows  that  35  is  to  be  divided 
by  7,  and  is  equivalent  to  35-*-7. 

OES.  It  will  be  perceived  that  division  is  similar  in  principle  to  sub- 
traction, and  may  be  performed  by  it.  For  instance,  to  find  how 
many  times  3  is  contained  in  12,  as  in  the  first  example,  subtract  3 
(the  divisor)  continually  from  12  (the  dividend)  until  the  latter  is  ex- 
hausted ;  then  counting  these  repeated  subtractions,  we  shall  have  the 
true  quotient.  Thus,  3  from  12  leaves  9 ;  3  from  9  leaves  6 ;  3  from 
6  leaves  3 ;  3  from  3  leaves  0.  Now  by  counting,  we  find  that  3  can 
be  taken  from  12,  4  times;  or  that  3  is  contained  in  12,  4  times. 
Hence, 

6  7  •  a.  Division  is  sometimes  defined  to  be  a  short  way 
of 'performing  repeated  subtractions  of  the  same  number. 

OBS.  1.  It  will  also  be  observed  that  division  is  the  reverse  of  mul- 
tiplication. Multiplication  is  the  repeated  addition  of  the  same  num- 
ber ;  division  is  the  repeated  subtraction,  of  the  same  number.  Tko 
product  of  the  one  answers  to  the  dividend  of  the  other :  but  the  lat- 
ter is  always  given,  while  the  former  is  required. 

2.  When  the  dividend  denotes  things  of  one  denomination  only,  the 
operation  is  called  Simple  Division. 

EXERCISES   FOR   THE   SLATE. 

Ex.  1.  How  many  barrels  of  cider,  at  2  dollars  a  bar. 
rel.  can  you  buy  for  648  dollars  ? 

Suggestlm. — Since  2  dollars  will  buy  1  barrel,  648  dol 
lars  will  buy  as  many  barrels  as  2  is  contained  times  in 
648. 


QUEST. — What  is  the  first  ?  What  does  this  sign  show  ?  What  ia 
the  second  way  of  denoting  division  ?  Obs.  To  what  rule  is  division 
similar  in  principle  ?  How  is  division  sometimes  defined  ?  Of  what 
is  division  the  reverse  ?  How  does  this  appear  ?  When  the  dividend 
denotes  things  of  one  denomination  only,  what  is  the  operation  called  ? 


ARTS.  68,  69.]  DIVISION.  79 

Having  written  the  numbers  upon  the      Operation. 
slate,  as  in  the  margin,  we  proceed  thus :  Divig0r.  Dividend. 
2  is  contained  in  6,  3  times.     Now  as  the         2)648 
6  denotes  hundreds,  the  3  must  also  be       ^    '  f 

hundreds.  We  therefore  write  it  in  hun-  ^uot"  *  *  * 
dreds'  place  ;  that  is,  under  the  figure  which  we  are  di- 
viding. 2  in  4,  2  times.  Since  the  4  is  tens,  the  2  must 
also  be  tens,  and  we  write  it  in  tens'  place.  2  in  8, 4  times. 
The  8  is  units  ;  hence  the  4  must  be  units,  and  we  write 
it  in  units'  place.  The  answer  is  324  barrels. 

2.  Divide  63  by  7.  Ans.  9. 

3.  Divide  56  by  8.  •     4.  Divide  42  by  7. 
5.  Divide  54  by  9.  6.  Divide  72  by  8. 

7.  How  many  hats,  at  2  dollars  apiece,  can  be  bought 
for  468  dollars  ?     Ans.  234  hats. 

8.  How  many  sheep,  at  3  dollars  a  head,  can  be  bought 
for  369  dollars? 

9.  A  man  wishes   to  divide  248  acres  of  land  equally 
•etween  his  two  sons :  how  many  acres  will  each  receive  1 

10.  How  many  times  is  4  contained  in  488  ? 

68.  Hence,  when  the  divisor  contains  but  one  figure, 

Write  the  divisor  on  the  left,  hand  of  the  dividend  with 
a  curve  line  between  them  ;  then,  beginning  at  the  left  hand, 
divide  each  figure  of  the  dividend  by  the  divisor,  and  set 
each  quotient  figure  directly  under  the  figure  from  which  it 
arose. 

11.  A  farmer  bought  96  dollars  worth  of  dry  goods, 
and  agreed  to  pay  in  wood  at  3  dollars  a  cord :  how  many 
cords  will  it  take  to  pay  his  bill  ?  Ans.  32  cords. 

12.  In  963  feet,  how  many  yards  are  there,  allowing  3 
feet  to  a  yard  ? 

13.  Divide  63936  by  3.         14.  Divide  48848  by  4. 
15.  Divide  55555  by  5.         16.  Divide  2486286  by  2 

69.  When  the  divisor  is  not  contained  in  the  first 

QUEST. — 68.  How  do  you  write  the  numbers  for  division  ?  Where 
begin  to  divide  ?  Where  place  each  quotient  figure  ?  69  When  the 
divisor  is  not  contained  in  the  first  figure  of  the  dividend,  what  must 
he  done  ? 


80  DIVISION.  [SECT.  V, 

figure  of  the  dividend,  we  must  find  how  many  times  i1 
is  contained  in  the  first  two  figures. 

1  7.  How  many  hats,  at  3  dollars  apiece,  can  be  bough 
for  249  dollars  ? 

Operation.       Since  the   divisor  3,  is  not  contained  in 
3)249    ^  the  first  figure  of  the  dividend,  we  say 
3  is  in  24,  8  times,  and  write  the  8  under 


,  , 

B  ,    the  4      3  in  9j  3  times  AnSm  83  hats 

18.  Divide  124  by  4.  19.  Divide  366  by  6. 

20.  Divide  255  by  5.  21.  Divide  1248  by  4. 

22.  Divide  24693  by  3.  23.  Divide  4266  by  6. 

24.  Divide  35555  by  5.  25.  Divide  5677  by  7. 

26.  Divide  64888  by  8.  27.  Divide  8199  by  9. 

7  O.  After  dividing  any  figure  of  the  dividend,  if  there 
is  a  remainder,  prefix  it  mentally  to  the  next  figure  of  the 
dividend,  and  then  divide  this  number  as  before. 

Note.  —  To  prefix  means  to  place  before,  or  at  the  left  hand. 

28.  A  man  bought  741  acres  of  land,  which  he  divi- 
ded equally  among  his  3  sons  :  how  many  acres  did  each 
receive  ? 

Operation.  When  \ve  divide  7  by  3,  there  is  1 

3)741  remainder.     This  we  prefix  mentally 

A  -  9A7  to  me   next   %ure  °f  tne   dividend. 

We  then  say,  3  in  14,  4  times,  and  2 
over.  Prefixing  the  remainder  2  to  the  next  figure,  as  be 
fore,  we  say,  3  in  21,  7  times. 

29.  If  a  man  travel  at  the  rate  of  5  miles  an  hour,  how 
long  will  it  take  him  to  travel  345  miles  ?  Ans.  69  hours. 

30.  If  192  pounds  of  flour  were  equally  divided  among 
4  persons,  how  many  pounds  would  each  receive  ? 

31.  Divide  45690  by  6.         32.  Divide  52584  by  8. 
33.  Divide  81670  by  5.         34.  Divide  28296  by  9. 
35.  When  flour  is  6  dollars  a  barrel,  how  much  cas. 

be  bought  for  642  dollars  ? 

QUEST.  —  70.  If  there  is  a  remainder  after  dividing  a  figure,  of  th<? 
dividend,  what  must  be  done  with  it  ?  Note.  What  does  the  word 
prefix  mean  ?  Obs.  When  the  divisor  is  not  contained  in  a  figure  c£ 
the  dividend,  what  must  be  done  1 


ARTS.  70,  71.]  DIVISION.  81 

OBS.  In  this  example  the  divisor  is  not  contain-      Operation 
ed  once  in  the  tens'  figure  of  the  dividend ;  we         6)642 
must  therefore  write  a  cipher  in  the  quotient,  and 
prefix  the  4  to  the  next  figure  of  the  dividend,  as    Ans'  l°7  barrels. 
if  it  were  a  remainder.    We  then  say,  6  in  42,  7  times,  and  place  the 
7  under  the  2. 

36.  Divide  36060  by  6.         37.  Divide  49000  by  7. 
38.  Divide  45900  by  9.         39.  Divide  568000  by  8. 

40.  Allowing  5  yards  of  cloth  for  a  suit  of  clothes,  how 
nany  suits  can  be  made  from  1525  yards  ? 

Ans.  305  suits. 

41.  A  company  of  3  men  agree  to  pay  a  bill  of  321 
•iollars :  how  many  dollars  must  each  man  pay  ? 

42.  Divide  14350  by  7.         43.  Divide  30420  by  6. 
44.  Divide  25105  by  5.         45.  Divide  643240  by  8. 

46.  A  merchant  wished  to  divide  49  oranges  equally 
among  4  boys :  how  many  must  he  give  to  each  ? 

Operation.  After  giving  them  12  apiece,  it 

4Ug  will  be  seen  that  there  is  one  re- 

mainder, or  1  orange  left,  which 
Ans.  12-1  remainder.     is  not  diyided     ^ow  it  is  plain 

that  the  whole  dividend  must  be  divided,  in  order  to  ren- 
der the  division  complete.  But  4  is  not  contained  in  1 ; 
hence  the  division  must  be  represented  by  writing  the  4 
under  the  1,  thus  £,  (Art.  67,)  and  in  order  to  complete 
the  quotient,  the  £  must  be  annexed  to  the  12.  The 
irue  quotient,  therefore,  is  12  and  1  divided  by  4,  and 
should  be  written  thus,  12£.  Hence, 

71,  When  there  is  a  remainder •,  after  dividing  the  last 
figure  of  the  dividend,  it  should  always  be  written  over  the 
divisor  and  annexed  to  the  quotient. 

47.  A  shoemaker  has  375  pair  of  boot?,  which   he 
wishes  to  pack  in  6  boxes :  how  many  pair  can  he  put 
into  a  box  ?  Ans.  62|. 

48.  A  baker  wishes  to  lay  out  756  dollars  in  flour : 
how  much  can  he  buy,  when  the  price  is  5  dollars  a  bar- 
rel? 


QUEST. — 71.  When  there  is  a  remainder   after  dividing  the  last 
Igure  of  the  dividend,  what  must  be  done  with  it  ? 


82  DIVISION.  [SECT.  V 

49.  How  many  yearlings,  at  9  dollars  a  head,  can  ba 
oougnt  for  468  dollars  ? 

50.  How  many  acres  of  land,  at  6  dollars  an  acre,  can 
1  buy  for  973  dollars  ? 

72.  The  preceding-  method  of  dividing  is  called  Sho-r 
Division.  From  the  illustrations  and  principles  now  ex- 
plained, we  derive  the  following 

RULE  FOR  SHORT  DIVISION. 

I.  Write  the  divisor  071  the.  left  hand  of  the  dividend  with 
a  curve  line  between  them.      Then  beginning  at  the  left  hand, 
divide  successively  each  figure  of  the  dividend  by  the  divisor, 
and  place  each  quotient  figure  directly  under  the  figure  di- 
vided. (Art.  68.) 

II.  If  there  is  a  remainder  after  dividing  any  figure, 
prefix  it  to  tlie  next  figure  of  the  dividend  and  divide  this 
number  as  before  ;  and  if  the  divisor  is  not  contained  in  any 
figure  of  the  dividend,  place  a  cipher  in  the  quotient,  and 
prefix  this  figure  to  the  next  one  of  the  dividend,  as  if  it  were 
a  remainder.  (Arts.  69,  70.  Obs.) 

III.  When  a  remainder  occurs  after  dividing   the   last 
figure,  write  it  over  the  divisor  and  annex  it  to  the  quotient. 
(Art  71.) 

7  3»  PROOF. — Multiply  the  divisor  by  the  quotient,  to  tfw 
product  add  the  remainder,  and  if  the  sum  is  equal  to  tlie, 
dividend,  the  work  is  right. 

51.  Divide  6973  by  6. 

Solution.  Pro°f'  H62  Quotient 

6  divisor. 
6)6973  6972 

Quot.    1162andlrm.         Add  the  rem.       1 

The  result  is  6973,  the  div. 


QUEST. — 72.  What  is  the  rule  for  short  division  ?  73.  How  is  di 
vision  proved  ?  Obs.  How  does  it  appear  that  the  product  of  the  di- 
visor and  quotient  should  be  equal  to  the  dividend  ?  What  other  waj 
of  proving  division  ia  mentioned  ? 


ARTS.  72-74.]  DIVISION.  83 

OBS.  1.  Since  the  quotient  shows  how  many  times  the  divisor  is 
contained  in  the  dividend,  (Art.  64,)  it  follows,  that  if  the  divisor 
is  repeated  as  many  times  as  there  are  units  in  the  quotient,  it  must 
produce  the  dividend. 

2.  Division  may  also  be  proved  by  subtracting  the  remainder,  if 
any,  from  the  dividend,  then  dividing  the  result  by  the  quotient. 

PROOF    OF   MULTIPLICATION   BY   DIVISION. 

74,  Divide  the  product  by  one  of  tJie  factors,  and  if  the 
quotient  thus  arising  is  equal  to  the  other  factor,  the  work 


OBS.  This  method  of  proof  depends  on  this  obvious  principle,  viz ; 
if  the  divisor  and  quotient,  multiplied  together,  produce  the  dividend, 
the  product  of  the  two  numbers,  divided  by  one  of  those  numbers, 
iiust  give  the  other  number. 

LONG  DIVISION. 

Ex.  1.  A  father  bought  741  acres  of  land,  which  he 
divided  equally  among  his  3  sons:  how  many  acres  did 
each  receive  ? 

Note. — This  ex:impi>'  IIAH  hrn  solved  by  short  division.  (Art.  70. 
Ex.  28.)  We  have  hitnxkuvd  it  here  for  the  purpose  of  illustrating 
a  different  mode  of  dividing. 

Having  written  the  divisor  on  the          Operation. 
luft  of  the  dividend  as  before,  we  find    Divisor.  Divid.  Quot. 
;)  is  contained  in  7,  2  times,  and  place         3)    741    (247 
the  2  on  the  right  of  the  dividend,  5' 

with  a  curve  line  between  them.    We  -r\ 

next  multiply  the  (''visor  bv  this  quo-  .^ 

lient  figure — 2  times  3  are  6 — and, 
placing  the  product  under  the  7,  the 
figure  divided,  subtract  it  therefrom. 
We  now  bring  down  the  next  figure 
of  the  dividend,  and  placing  it  on  the  right  of  the  remain- 
der I,  we  have  14.  And  3  is  in  14,  4  times.  Set  the  4 


QUEST. — 74.  How  is  multiplication  proved  by  division  ?  Ols.  Upon 
what  principle  does  this  proof  depend  ?  How  are  the  numbers  written 
for  long  division  ?  Where  begin  to  divide  ?  Where  is  the  quotient 
placed  1 


84  DIVISION.  [SECT.  \ 

on  the  right  hand  of  the  last  quotient  figure,  and  multi 
ply  the  divisor  by  it:  4  times  3  are  12.  Write  the  pro- 
duct under  14,  and  subtract  as  before.  Finally,  bringing 
down  the  last  figure  of  the  dividend  to  the  right  of  the 
last  remainder,  we  have  21 ;  and  3  is  in  21,  7  times. 
Set  the  7  in  the  quotient,  then  multiply  and  subtract  as 
before.  The  quotient  is  247,  the  same  as  in  short  division. 

75.  This  method  of  dividing  is  called  Long  Division. 
It  is  the  same  in  principle  as  Short  Division.  The  only 
difference  between  them  is,  that  in  Long  Division  the 
result  o'f  each  step  in  the  operation  is  written  down,  while 
in  Short  Division  we  carry  on  the  process  in  the  mind, 
and  simply  write  the  quotient. 

Note. — To  prevent  mistakes,  it  is  advisable  to  put  a  dot  under 
each  figure  of  the  dividend,  when  it  is  brought  down. 

The  following  questions  are  designed  to  be  performed  by  long 
division,  and  each  operation  should  be  proved. 

2.  How  many  times  is  2  contained  in  578  ?  Ans.  289. 

3.  How  many  times  is  5  contained  in  7560  ? 

Ans.  1512. 

4.  How  many  times  is  4  contained  in  126332  ? 

Ans.  31583. 

5.  How  many  times  is  6  contained  in  763251  ? 

6.  How  many  times  is  3  contained  in  4026942  ? 

7.  How  many  times  is  8  contained  in  2612488? 

8.  How  many  times  is  5  contained  in  1682840? 

9.  How  many  times  is  7  contained  in  45063284  ? 

10.  How  many  times  is  9  contained  in  650031507? 

11.  Divide  2234  by  21. 

Operation,  21  is  contained    in  22  once. — 

21)2234(106-8-.  Ans.     Write  the  1  in  the  quotient.    Then 

2i  multiplying  and  subtracting,  the 

-r^7  remainder  is  1.     Bringing  down 

the  next  figure,  we  have  1 3  to  be 

divided  by  21.     But  21  is  not  con- 

8  rem-  tained  in   13,  therefore  we  put  a 


QUEST. — 75.  What  is  the  difference  between  long  and  short  divisio*.  ' 


ARTS.  75-77. J  DIVISION.  85 

cipher  in  the  quotient,  (Art.  70.  Obs.)  and  bring1  down 
the  next  figure.  Then,  21  in  134, 6  times,  and  8  remain- 
der. Write  the  8  over  the  divisor,  and  annex  it  to  the 
quotient.  (Art.  71.) 

76.  After  the  first  quotient  figure  is  obtained,  formed 
figure  of  the  dividend  which  is  brought  down,  either  a  sig- 
nificant figure  or  a  cipher  must  be  put  in  the  quotient. 

12.  Divide  345  by  15.  Ans.  23. 

13.  Divide  5378  by  25.  Ans.  215  JL. 

14.  Divide  7840  by  32.  16.  Divide  59690  by  45. 
1G.  Divide  81229  by  67.  17.  Divide  99435  by  81. 

18.  How  many  times  is  131  contained  in  18602? 

Ans.  142. 

OBS.  When  the  divisor  is  not  contained  in  the  first  two  figures  of 
the  dividend,  find  how  many  times  it  is  contained  in  the  first  three ; 
and,  generally,  find  how  many  times  it  is  contained  in  the  fewest  fig- 
ures which  will  contain  it,  and  proceed  as  before. 

19.  How  many  times  is  93  contained  in  100469  ? 

20.  How  many  times  is  156  contained  in  140672? 

77.  From  the  preceding  principles  we  derive  the  fol- 
lowing 

RULE  FOR  LONG  DIVISION. 

Begin  on  the  left  of  the  dividend,  find  how.  many  times 
the  divisor  is  contained  in  the  fewest  figures  thai,  will  con- 
tain it,  and  'place  the  quotient  figure  on  the  right  of  the 
dividend  with  a  curve  line  between  tJiem.  Then  multiply  the 
divisor  by  this  figure  and  subtract  the  product  from  the  fig- 
ures divided ;  to  the  right  of  the  remainder  bring  down  the 
next  figure  of  the  dividend  and  divide  this  number  as  before. 
Proceed  in  this  manner  till  all  the  figures  of  the  dividend  are 
diviflcd. 

When  there  is  a  remainder  after  dividing  the  last  figure, 
write  it  over  the  divisor  and  annex  it  to  the  quotient,  as  in 
thort  division.  (Art.  71.) 

QUEST. — 76.  What  is  placed  in  the  quotient,  on  bringing  down  each 
figure  of  the  dividend  ?  Obs.  When  the  divisor  is  not  contained  in 
the  first  two  figures  of  the  dividend,  what  is  to  be  dune  ?  77.  What  is 
the  rule  for  long  division  ? 


86  DIVISION.  [SECT.  V. 

OBS.  When  the  divisor  contains?  but  one  figure,  the  operation  by 
Short  Division  is  the  most  expeditious,  and  should  therefore  be  prac- 
ticed ;  but  when  the  divisor  contains  two  or  more  figures,  it  will  ge- 
nerally be  the  most  convenient  to  divide  by  Long  Division, 

EXAMPLES   FOR   PRACTICE. 

1.  If  a  man  travel  at  the  rate  of  8  miles  an  hour,  how 
long  will  it  take  him  to  travel  192  miles? 

2.  How  many  yards  of  broadcloth,  at  9  dollars  a  yard, 
•;an  be  bought  for  324  dollars  1 

3.  A  farmer  bought  a  lot  of  young  cattle,  at  1 1  dollars 
per  head,  and  paid  473  dollars  for  them :  how  many  did 
he  buy  1 

4.  How  many  tons  of  coal,  at  7  dollars  a  ton,  can  be 
bought  for  756  dollars  ? 

5.  At  12  dollars  a  month,  how  long  will  it  take  a  man 
to  earn  156  dollars? 

6.  In  one  day  there  are  24  hours :  how  many  days  are 
there  in  480  hours  ? 

7.  A  man  traveled  215  miles  in  21  hours :  how  many 
miles  did  he  travel  per  hour  ? 

8.  At  16  dollars  a  ton,  how  many  tons  of  hay  can  be 
bought  for  176  dollars? 

9.  How  many  casks  of  wine,  at  25  dollars  a  cask,  can 
be  bought  for  275  dollars  ? 

10.  The  ship  George  Washington  was  25  days  in  cross- 
ing the  Atlantic  Ocean,  a  distance  of  3000  miles.     How 
many  miles  did  the  ship  sail  per  day  ? 

11.  The  steamer  Great  Western  crossed  it  in  15  days. 
How  many  miles  did  she  sail  per  day  ? 

12.  The  steamer  Caledonia  crossed  it  in  12  days.    How 
many  miles  did  she  sail  per  day? 

13.  If  a  man  can  earn  32  dollars  a  month,  how  long 
will  it  take  him  to  earn  420  dollars  ? 

14.  If  63  gallons  make  a  hogshead,  how  many  hogs- 
heads  will  1260  gallons  make? 

15.  If  a  ship  can  sail  264  miles  per  day,  how  far  can 
"she  sail  in  an  hour? 

QURST. — Obs.  When  should  short  division  be  used*    When  long 
division  ? 


ART.  77.  a.]  DIVISION.  87 

16.  How  many  tiriles  12  in  172,  and  how  many  over  ? 

17.  How  many  times  15  in  630,  and  how  many  over? 

18.  How  many  times  22  in  865,  and  how  many  over? 

19.  1236  is  how  many  times  17,  and  how  many  over  ? 

20.  7652  is  how  many  times  13,  and  how  many  over? 

21.  3061  is  how  many  times  125,  and  how  many  over? 

22.  1861  is  how  many  times  231,  and  how  many  over? 

23.  8  times  256  is  how  many  times  9  ? 

24.  12  times  157  is  how  many  times  7? 

25.  15  times  2251  is  how  many  times  12  ? 

26.  19  times  136  is  how  many  times  75  ? 

27.  63  times  102  is  how  many  times  37  ? 

28.  78  times  276  is  how  many  times  136? 

29.  115  times  321  is  how  many  times  95? 

30.  144  times  137  is  how  many  times  312? 

CONTRACTIONS  IN  DIVISION. 

7  7  •  a.  The  operations  in  division,  as  well  as  in  mul 
implication,  may  often  be  shortened  by  a  careful  attention 
,o  the  application  of  the  preceding  principles. 

CASE  I. —  When  the.  divisor  is  a  composite  number. 

Ex.  1.  A  gentleman  divided  168  oranges  equally 
among  14  grandchildren  who  belonged  to  2  families, 
each  family  containing  7  children :  how  many  oranges  did 
he  give  to  each  child  ? 

Suggestion. — First  find  how  many  each  family  received, 
I  hen  how  many  each  child  received. 

If  2  families  receive  168  oranges,  1  fami-  ^ 

.  y  will  receive  as  many  oranges,  as  2  is  ?er 

contained  times  in  168,  viz:  84.    But  there  2)168 

are  7  children  in  each  family.     If  then  7  7)84 

children  receive  84  oranges,  1  child  will  ~^  Ans 
receive  as  many,  as  7  is  contained  times  in 
84,  viz :  12.     He  therefore  gave  12  oranges  to  each  child. 

NOTE. — This  operation  is  exactly  the  reverse  of  that  in  Ex.  1. 
Art.  55.  The  divisor  14  being  a  composite  number,  we  divide  first 
by  one  of  its  factors,  and  the  quotient  thus  found  by  the  other.  The 
final  result  would  have  been  the  same,  if  we  had  divided  by  7  first, 
then  by  3.  Hence, 


88  DIVISION.  [SECT.  V, 

78.  To  divide  by  a  composite  riUmber. 

Divide  the  dividend  by  one  of  the  factors  of  the  divisor 
find  the  quotient  thus  obtained  by  the  other  factor.  The.  last 
quotient  will  be  the  answer  required. 

To  find  the  true  remainder,  should  there  he  any. 
Multiply  the  last  remainder  by  the  first  divisor,  and  to  the. 
product  add  the  first  remainder. 

OBS.  1.  If  the  divisor  can  be  resolved  into  more  than  two  factors, 
we  may  divide  by  them  successively,  as  above. 

*2.  To  find  the  true  remainder  when  more  than  two  factors  are  em- 
ployed, multiply  each  remainder  by  all  the  preceding  divisors,  and 
to  the  sum  of  the  products  add  the  iirst  remainder. 

2.  Divide  465  by  35. 

1  last  remainder. 
Tfirstdiviso, 


-         0  7  product. 

J*  first  rem.  added. 
10  true  rem.  Ans. 


3.  A  teacher  having  36  scholars  arranged  in  4  equal 
classes,  wishes  to  distribute  216  pears  among  thein  equally  : 
how  many  can  he  give  to  each  scholar  ? 

4.  How  many  cows,  at  27  dollars  a  head,  can  be  bought 
for  945  dollars  'I 

5.  How  many  times  is  64  contained  in  453  ? 

6.  How  many  times  is-  72  contained  in  237  ? 

CASE  II.  —  When  the  divisor  is  \  iwth  ciphers  annexed 
to  it. 

7  9.  It  has  been  shown  that  annexing  a  cipher  to  a 
number,  increases  its  value  ten  times,  or  multiplies  it  by 
10.  (Art.  58.)  Reversing  this  process;  that  is,  remo 
ring  a  cipher  from  the  right  hand  of  a  number,  will  evi- 
dently diminish  its  value  ten  times,  or  divide  it  by  10  ;  for, 

QUEST.  —  78.  How  proceed  when  the  divisor  is  a  composite  number  1 
How  find  the  true  remainder  I  Ols.  How  proceed  when  the  divisor 
can  be  resolved  into  more  than  two  factors  ?  How  find  the  remaindei 
in  this  case  ?  79.  What  is  the  effect  of  annexing  a  cipher  to  a  num- 
ber ?  What  is  the  effect  of  removing  a  cipher  from  the  right  of  a 
number  ? 


ARTS.  78-80.]  DIVISION.  89 

each  figure  in  the  number  is  thus  restored  to  its  original 
place,  and  consequently  to  its  original  value.  Thus,  an- 
nexing a  cipher  to  12,  it  becomes  120,  which  is  the  same 
as  12x10.  On  the  other  hand,  removing  the  cipher  from 
120,  it  becomes  12,  which  is  the  same  as  120-*-10. 

In  the  same  manner  it  may  be  shown,  that  removing 
two  ciphers  from  the  right  of  a  number,  divides  it  by  100 
removing  three,  divides  it  by  1000  ;  removing  four,  di 
vides  it  by  10000,  &c.  Hence, 

8O.  To  divide  by  10,  100,  1000,  &c. 

• 

Cut  of  as  many  figures  from  the  right  hand  of  the  divi- 
dend as  there  are  ciphers  in  the  divisor.  The  remaining 
figures  of  tlie  dividend  will  be  the  quotient,  and  those  cut  off 
the  remainder. 

7.  How  many  times  is  10  contained  in  120? 

Ans.  12. 

8.  In  one  dime  there  are  10  cents :  how  many  dimes 
are  there  in  100  cents?     In  250  cents?     In  380  cents? 

9.  In  one  dollar  there  are  100  cents:  how  many  dol- 
lars are  there  in  6500  cents?      In  76500  cents?     In 
432000  cents  ? 

10.  Divide  675000  by  10000. 

Ans.  67  and  5000  rem. 

11.  Divide  44360791  by  1000000. 

12.  Divide  82367180309  by  10000000. 

CASE  III. —  When  the  divisor  has  ciphers  on  the  right. 

13.  How  many  acres  of  land,  at  20  dollars  per  acre, 
can  you  buy  for  645  dollars  ? 

Analysis. — The  divisor  20  is  a  composite  number,  the 
factors  of  which  are  2  and  10.  (Art.  55.  Obs.  1.)  We 
may,  therefore,  divide  first  by  one  factor,  and  the  quo- 
tient thence  arising  by  the  other.  (Art.  78.)  Now 
cutting  off  the  right  hand  figure  of  the  dividend,  divides 
it  by  10 ;  (Art.  80 ;)  consequently,  dividing  the  remaining 

QWEST.— 80.  How  proceed  when  the  divisor  is  10,  100,  1000,  &c. 


*0  DIVISION.  [SECT.  V 

figures  of  the  dividend  by  2,  the  other  factor  of  the  di 
visor,  will  give  the  true  quotient. 

sy       .  •  Cut  off  the  cipher  on  the  right  of  the 

divisor ;  also  cut  off  the  right  hand  figure 

of  the  dividend ;  then  divide  the  64  by 

—  2.     The  5  which  we  cut  off,  is  the  re- 

32-5  rem.     mainder.     Ans.  32/7  acres.     Hence, 

8 1 .  When  there  are  ciphers  on  the  right  hand  of  the 
divisor. 

Cut  off  the  ciphers,  also  cut  off  as  many  figures  from 
the  right  of  the  dividend.  Then  divide  the  other  figures 
of  the  dividend  by  t/ie  remaining  figures  of  the  divisor, 
and  annex  the  figures  cut  off  from  the  dividend  to  the  re- 


14.  How  many  horses,  at  80  dollars  apiece,  can  you 
buy  for  640  dollars  ? 

15.  How  many  barrels  will  6800  pounds  of  beef  make, 
allowing  200  pounds  to  the  barrel? 

16.  How  many  regiments  of  4000  each,  can  be  formed 
from  840000? 

17.  Divide  143900  by  2100. 

18.  Divide  4314670  by  24000. 

8 1  •  a.  The  four  preceding  rules,  viz :  Addition,  Sub- 
traction, Multiplication,  and  Division,  are  usually  called 
the  FUNDAMENTAL  RULES  of  Arithmetic,  because  they  are 
the  foundation  or  basis  of  all  arithmetical  calculations. 

GENERAL  PRINCIPLES  IN  DIVISION. 

82.  From  the  nature  of  division,  it  is  evident,  that  the 
value  of  the  quotient  depends  both  on  the  divisor  and  the 
dividend. 

If  a  given  divisor  is  contained  in  a  given  dividend  a 

QUEST. — 81.  When  there  are  ciphers  on  the  right  of  the  divisor,  how 
proceed  ?  What  is  to  be  done  with  figures  cut  off  from  the  dividend  ! 
81.  a.  What  are  the  four  preceding  rules  called?  Why?  82.  Upon 
what  does  the  value  of  the  quotient  depend  ? 


ARTS.  81-85.]  DIVISION.  91 

certain  number  of  times,  the  same  divisor  will  obviously 

be  contained, 

In  double  that  dividend,  twice  as  many  times ;    • 

In  three  times  tir-it  dividend,  thrice  as  many  times;  &c. 

Thus,  4  is  contained  in  12,  3  times;  in  2  times  12  or 

24,  4  is  contained  6  times ;  (i.  e.  twice  3  times ;)  in  3 

times   12  or  36,  4  is  contained  9  times;  (i.  e.  thrice  3 

times ;)  &c.     Hence, 

83»  If  the  divisor  remains  the  same,  multiplying  the 
dividend  by  any  number,  is  in  effect  multiplying  the  quotient 
by  that  number. 

Again,  if  a  given  divisor  is  contained  in  a  given  divi- 
dend a  certain  number  of  times,  the  same  divisor  is  con- 
tained, 

In  half  that  dividend,  half  as  many  times  ; 

In  a  third  of  that  dividend,  a  third  as  many  times,  &c. 

Thus,  4  is  contained  in  24,  6  times ;  in  24-*-2  or  12, 
(rnlf  of  24,)  4  is  contained  3  times  ;  (i.  e.  half  of  6  times  ;) 
in  24-*-3  or  8,  (a  third  of  24,)  4  is  contained  2  times  ; 
(i.  e.  a  third  of  6  times ;)  &c.  Hence, 

£4--  If  the  divisor  remains  the  same,  dividing  the  divi- 
dend by  any  number,  is  in  effect  dividing  the.  quotient  by  that 
number. 

If  a  given  divisor  is  contained  in  a  given  dividend  a 
certain  number  of  times,  then,  in  the  same  dividend, 

Twice  that  divisor  is  contained  only  half  as  many  times ; 

Three  times  that  divisor,  a  third  as  many  times,  &c. 

Thus,  2  is  contained  in  12,  6  times ;  2  times  2  or  4,  is 
contained  in  12,  3  times;  (i.  e.  half  of  6  times ;)  3  times 
2  or  6,  is  contained  in  12,  2  times  ;  (i.  e.  a  third  of  6 
times ;)  &c.  Hence, 

85.  If  the  dividend  remains  the  same,  multiplying  the 
divisor  by  any  number,  is  in  effect  dividing  the  quotient  by 
tJiat  number. 

QUEST. — 83.  If  the  divisor  remains  the  same,  what  effect  has  it  on 
the  quotient  to  multiply  the  dividend?  84.  What  is  the  effect  of  divi- 
ding the  dividend  by  any  given  number  ?  85.  If  the  dividend  remains 
the  same,  what  is  the  effect  of  multiplying;  the  divisor  by  any  given 
number  ? 


92  DIVISION.  [SECT.  V 

If  a  given  divisor  is  contained  in  a  giver  dividend  a 
certain  number  of  times,  then,  in  the  same  dividend, 
Half^  that  divisor  is  contained  twice  as  many  times ; 
A  third  of  that  divisor,  three  times  a?  many  times,  &c. 

Thus,  6  is  contained  in  24,  4  times :  6-*-2  or  3,  (half 
of  6,)  is  contained  in  24,  8  times ;  (i.  e.  twice  4  times ;) 
6-*-3  or  2,  (a  third  of  6,)  is  contained  in  24,  12  times ;  (i.  e. 
three  times  4  times  ;)  &c.  Hence, 

86»  If  the  dividend  remains  the  same,  dividing  the  di- 
visor by  any  number,  is  in  effect  multiplying  the  quotient  by 
that  number. 

87.  From  the  preceding  articles,  it  is  evident  that  any 
given  divisor  is  contained  in  any  given  dividend,  just  a? 
many  times,  as  twice  that  divisor  is  contained  in  twice  thai 
dividend ;  three  times  that  divisor  in  three  times  that  div- 
idend, &c. 

Conversely,  any  given  divisor  is  contained  in  any  given 
dividend  just  as  many  times,  as  half  that  divisor  is  con- 
tained in  half  that  dividend  ;  a  third  of  that  divisor,  in  a 
third  of  that  "dividend,  &c. 

Thus,  4  is  contained  in  12,  3  times ; 

2  times  4  is  contained  in  2  times  12,  3  times ; 

3  times  4  is  contained  in  3  times  12,  3  times,  &c. 

Again,  6  is  contained  in  24,  4  times ; 
6-^-2  is  contained  in  24-J-2,  4  times ; 
6-s-3  is  contained  in  24-5-3,  4  times,  &c.     Hence 

88.  If  the  divisor  and  dividend  are  both  multiplied, 
or  both  divided  by  the  same  number,  the  quotient  will  not  bt 
altered. 

89.  If  any  given  number  is  multiplied  and  the  product 
divided  by  the  same  number,  its  value  will  not  be   altered. 
rnus,  12x5=60;  and  60-7-5=12,  the  given  number. 

QUEST.— 86.  What  of  dividing  the  divisor  ?  88.  What  is  the  effect 
upon  the  quotient  if  the  divisor  and  dividend  are  both  multiplied  or 
both  divided  by  the  same  number  \  89.  What  is  the  effect  of  m«lli- 
plying  and  dividing  any  given  number  by  the  same  number  ? 


ARTS.  86-91.]  DIVISION.  93 

CANCELATION.* 

90.  We  have  seen  that  division  is  finding  a  quotient, 
which  multiplied  into  the  divisor  will  produce  the  divi- 
dend.    (Art.  65.)     If,  therefore,  the  dividend  is  resolved 
into  two  such  factors  that  one  of  them  is  the  divisor,  the 
other  factor  will,  of  course,  be  the  quotient.     Suppose,  for 
example,  42  is  divided  by  6.     Now  the  factors  of  42  are 
6  and  7,  the  first  of  which  being  the  divisor,  the  other 
must  be  the  quotient.     Therefore, 

Canceling  a  factor  of  any  number,  divides  the  number  by 
that  factor.  Hence, 

91.  When  the  dividend  is  the  product  of  two  or  more 
factors,  one  of  which  is  the  same  as  the  divisor,  the  division 
way  be  performed  by  CANCELING  that  factor  in  the  divisor 
and  dividend.     (Art.  88.) 

Note. — The  term  cancel,  means  to  erase  or  reject. 

21.  Divide  the  product  of  19  into  25  by  19. 
Common  Method. 

19 

ne  By  Cancelation. 

^  10)10X25 

38  25  Ans. 

io\A7^/9c;    A  Cancel  the  factor  19,  which  is  com 

38  mon  both  to  the  divisor  and  dividend> 

and  25,  the  other  factor  of  the  dividend, 

is  the  quotient.     (Art  90.) 
y  o 

22.  Divide  85x31  by  85.     Ans.  31. 

23.  Divide  76x58  by  58. 

24.  Divide  75x40  by  40. 

25.  Divide  63x28  by  7. 

Analysis. — 28=4x7.  We  may  therefore  contract  the 
division  by  canceling  the  7,  which  is  a  factor  both  of  the 
dividend  and  the  divisor.  (Arts.  88,  90.) 

QUEST.— 90.  What  is  the  effect  of  canceling  a  factor  of  any  number  ? 
Note.  What  is  meant  by  the  term  cancel  ?  91.  When  the  divisor  is  a 
factor  of  the  dividend,  how  may  the  division  be  performed  ? 

*  Birk'a  Arithmetical  Collections,  London,  1 764. 


94 


DIVISION.  [SECT. 


Operation. 

#)63x4x#  The  product  of  63x4,  the  other  factors  of 

252  Ans.      tne  dividend,  is  the  answer  required. 

26.  In  32  times  84,  how  many  times  8  1     Ans.  336. 

27.  In  35  times  95,  how  many  times  7  ? 

28.  In  48  times  133,  how  many  times  8  ? 

29.  In  96  times  156,  how  many  times  12? 

30.  Divide  168x2x7  by  7x3. 

Operation. 

/?X3)168x2xff         We  cancel  the  factor  7,  which  is  corn- 

3)336  mon  to  the  divisor  and  dividend,  then 

112  Ans.  divide  the  product  of  168  into  2  by  3. 

31.  Divide  the  product  of  8,  6,  and  12  by  the  product 
of  2,  6,  and  8. 


Solution.  —  2x0X$)$X6x-*2=6.  Ans. 

Note.  —  We  cancel  the  factors  2,  6  and  8  in  the  divisor,  and  the  12 
and  8  in  the  dividend.  Canceling  the  same  or  equal  factors,  both  in 
the  divisor  and  dividend,  is  dividing  them  both  by  the  same  number, 
and  consequently  does  not  affect  the  quotient.  (Arts.  88,  90.)  Hence, 

91.  a.    When  the  divisor  and  dividend  have  factors 
common  to  both,  the  division  may  be  performed  by  canceling 
the  common  factors,  and  then  dividing  those  that  are  left  as 
before. 

32.  Divide  the  product  of  7,  9,  15,  and  8  by  the  pro- 
duct of  5,  7,  and  8. 

33.  Divide  the  product  of  6,  3,  7,  and  4  by  the  product 
of  12  and  6. 

34.  Divide  the  product  of  2,  28,  and  15  by  30. 

35.  Divide  the  product  of  5,  6,  and  56  by  7x3. 

92.  The   method  of  contracting  arithmetical  opera- 
tions, by  rejecting  equal  factors,  is  called  CANCELATION.    It 
applies  with  great  advantage  to  that  class  of  examples  and 
problems  which  involve  both  multiplication  and  division  ; 
that  is,  when  the  product  of  two  or  more  numbers  is  to  be 
divided  by  another  number,  or  by  the  product  of  two  or 
more  numbers. 

Note.  —  Its  farther  developments  and  application  may  be  seen  in  re- 


ARTS.  91.  a.-94]  DIVISION.  95 

d  action  of  compound  fractions  to  simple  ones ;  in  multiplication  and 
division  of  fractions ;  in  simple  and  compound  proportion,  &c.,  &«. 

GREATEST  COMMON  DIVISOR. 

92.  a.  A  Common  Divisor  of  two  or  more  numbers,  is 
a  number  which  will  divide  them  without  a  remainder. 
Thus,  2  is  a  common  divisor  of  4,  6,  8,  12,  16. 

93.  The  Greatest  Common  Divisor  of  two  or  more 
numbers,  is  the  greatest  number  which  will  divide  them 
without  a  remainder.     Thus,  6  is  the  greatest  common 
divisor  of  12,  18,  and  24. 

OBS.  1.  One  number  is  said  to  be  a  measure  of  another,  when  the 
former  is  contained  in  the  latter  any  number  of  times  without  a  re- 
mainder. Hence,  a  Com.  divisor  is  often  called  a  Common  Measure. 

2.  It  will  be  seen  that  a  common  divisor  of  two  or  more  numbers,  is 
simply  a  factor  which  is  common  to  those  numbers,  and  the  greatest 
common  divisor  is  the  greatest  factor  common  to  them.  Hence, 

94.  To  find  a  common  divisor  of  two  or  more  num- 
bers. 

Resolve  each  number  into  two  or  more  factor s,  one  of  which 
shall  be  common  to  all  the  given  numbers. 

Ex.  1.     Find  a  common  divisor  of  8,  10,  and  12. 

Analysis. — 8  may  be  resolved  into  the  factors  2  and  4 ; 
that  is,  8=2x4  ;  10=2x5  ;  and  12=2x6.  Now  the  fac- 
tor 2  is  common  to  each  number  and  is  therelbre  a  com- 
mon divisor  of  them. 

2.  Find  a  common  divisor  of  9,  15,  18,  and  24. 

OBS.  The  following  facts  may  assist  the  learner  in  finding  common 
divisors  : 

1.  Any  number  ending  in  0,  or  an  even  number,  as  2,  4,  6,  &c. 
may  be  divided  by  2. 

2.  Any  number  ending  in  5  or  0,  may  be  divided  by  5. 

3.  Any  number  ending  in  0,  may  be  divided  by  10. 

4.  When  the  two  right  hand  figures  are  divisible  by  4,  the  whole 
number  may  by  divided  by  4. 

3.  Find  a  common  divisor  of  16,  20,  and  36. 

QUEST. — 92.  a.  What  is  a  common  divisor  of  two  or  more  numbers  ? 
93.  What  is  the  greatest  common  divisor  of  two  or  more  numbers? 
Obs.  When  is  one  number  said  to  be  a  measure  of  another  ?  What 
is  a  common  divisor  sometimes  called  ?  94.  How  do  you  find  a  com- 
mon divisor  of  two  or  more  numbers  ? 


DIVISION.  [SECT.  V, 

4.  Find  a  common  divisor  of  35,  50,  75,  and  80. 

5.  Find  a  common  divisor  of  148  and  184. 

6.  Find  a  common  divisor  of  126  and  4653. 

95.  No  two  numbers  can  have  a  common  divisor 
greater  than  a  unit,  unless  they  have  a  common  factor 
Thus,  the  factors  of  8  are  2  and  4 ;  the  factors  of  15  an 
3  and  5  ;  hence,  8  and  15  have  no  common  divisor. 

96.  To  find  the  greatest  common  divisor  of  two  num 
bers. 

Divide  the  greater  number  by  the  less;  then  the  prece- 
ding divisor  by  the  last  remainder,  and  so  on,  till  nothing 
remains.  The  last  divisor  ivill  be  the  greatest  common  di- 
visor. 

7.  What  is  the  greatest  common  divisor  of  70  and  84  ? 
Operation.  Dividing  84  by  70,  the  remainder  is  14  ; 
70)84(1             then  dividing  70  (the  preceding  divisor) 

70  by  14,  (the  last  remainder,)  nothing    re- 

T4\70(5       mains.     Hence,  14  the  last  divisor,  is  the 
70  greatest  common  divisor. 

8.  What  is  the  greatest  common  divisor  of  63  and  147 1 

9.  What  is  the  greatest  common  divisor  of  91  and  117? 

10.  What  is  the  greatest  common  divisor  of  247  and 
323? 

11.  What  is  the  greatest  common  divisor  of  285  and 
465? 

12.  What  is  the  greatest  common  divisor  of  2145  and 
3471? 

97.  To  find  the  greatest  common  divisor  of  more 
than  two  numbers. 

First  find  the  greatest  common  divisor  of  any  two  of 
them;  then,  that  of  the  common  divisor  thus  obtained  and 
of  another  given  number,  and  so  on  through  all  the  given 

QUEST. — 95.  If  two  numbers  have  not  a  common  factor,  what  is  true 
as  to  a  common  divisor  ?  96.  plow  find  the  greatest  common  divisoi 
of  two  numbers  ?  97.  Of  more,  than  two  ? 


ARTS.  95-100.]  DIVISION.  97 

numbers.     The  last  common  divisor  found j  loiH  be  the  one  re- 


13.  What  is  the  greatest  common  divisor  of  63,  105, 
and  140?  Ans.7. 

Suggestion. — Find  "the  greatest  common  divisor  of  63 
and  105,  which  is  21.  Then,  that  of  21  and  140. 

14.  What  is  the   greatest  common  divisor  of  16,  24, 
rind  100? 

15.  What  is  the  greatest  common  divisor  of  492,  744, 
and  1044? 

LEAST  COMMON  MULTIPLE. 

98.  One  number  is  said  to  be  a  multiple  of  another 
when  the  former  can  be  divided  by  the  latter  without  a 
remainder.     Thus,  4  is  a  multiple  of  2;  10  is  a  mul- 
tiple of  5. 

OBS.  A  multiple  is  therefore  a  composite  number,  and  the  num- 
ber thus  contained  in  it,  is  always  one  of  its  factors. 

99.  A  common  multiple  of  two  or  more  numbers,  is 
a  number  which  can  be  divided  by  each  of  them  without 
a  remainder.     Thus,  12  is  a  common  multiple  of  2,  3, 
and  4  ;  15  is  a  common  multiple  of  3  and  5. 

OBS.  A  common  multiple  is  also  a  composite  number,  of  which 
each  of  the  given  numbers  must  be  a  factor ;  otherwise  it  could  not 
be  divided  by  them. 

100.  The  continued  product  of  two  or  more  given 
numbers,  will  always  form  a  common  multiple  of  those 
numbers. 

The  same  numbers,  therefore,  may  have  an  unlimited 
number  of  common  multiples  ;  for,  multiplying  their  con- 
tinued product  by  any  number,  will  form  a  new  common 
multiple.  (Art.  99.  Obs.) 


QUEST.— 98.  What  is  a  multiple  of  a  number  ?  Obs.  What  kind 
of  a  number  is  a  multiple  ?  99.  What  is  a  common  multiple  I  Obs. 
What  kind  of  a  number  is  a  common  multiple  ?  100.  How  may  a 
common  multiple  of  two  or  more  numbers  be  found  1  How  many  cow- 
mon  multiples  may  there  be  of  any  given  numbers  I 


98  DIVISION.  [SECT.  V, 

101.  The  hast  common  multiple  of  two  or  more  num- 
bers, is  the  least  number  which  can  be  divided  by  each 
of  them  without  a  remainder.     Thus,  12  is  the  least  com' 
mon  multiple  of  4  and  6,  for  it  is  the  least  number  which 
can  be  exactly  divided  by  them. 

15.  Find  the  least  common  multiple  of  6  and  10. 

Analysis. — 6=2x3  ;  and  10=2x5.  Now  it  is  evident 
that  the  number  required  must  contain  all  the  different 
factors  which  are  in  each  of  the  given  numbers ;  other- 
wise it  will  not  be  a  common  multiple  of  them.  (Art.  99. 
Obs.)  The  continued  product  of  the  factors  2x3x2x 
5=60,  is  exactly  divisible  by  6  and  10,  but  it  will  be  seen 
that  60  is  twice  as  large  as  is  necessary  to  be  a  common 
multiple  of  them.  We  also  perceive  that  the  factor  2  is 
common  to  both  the  given  numbers ;  hence  it  is  that  the 
continued  product  is  twice  too  large.  If,  therefore,  we 
retain  this  factor  only  once,  the  continued  product  of  2x3x 
5=30,  which  is  the  smallest  number  that  is  exactly  di- 
visible by  6  and  10,  and  is  therefore  the  least  common 
multiple  of  them. 

Operation.  We  divide  both  numbers  by  2.     This 

2\g   /;   ,Q        resolves  them  into  factors,  and  the  divisor 

'—±* and  quotients  contain  all  the  different  fac- 

3  "  5  tors  found  in  each  of  the  given  numbers 
2x3x5=30  once,  and  only  once.  Then  we  multiply  the 
divisor  ant!  quotients  together  and  the  pro  duct  is  30,whichis 
the  least  common  multiple  required.  Hence, 

102.  To  find  the  least  common  multiple  of  two  or 
more  numbers. 

Write  the  given  numbers  in  a  line  with  two  points  be- 
tween them.  Divide  by  the  smallest  number  which  will  di- 
vide any  two  or  more  of  them  without  a  remainder,  and  set 
the  quotients  and  the  numbers  not  divided  in  a  line  belmo. 
Divide  this  line  and  set  down  the  results  as  before;  thus 


QUEST. — 101.  What  is  the  least  common  multiple  of  two  or  more 
numbers  ?  102.  How  is  the  least  common  multiple  of  two  or  mor« 
numbers  found  ? 


ARTS.  "01,  102.1  DIVISION.  99 

continue  the  operation  till  tJiere  are  no  two  numbers  which 
can  be  divided  by  any  number  greater  than  1.  The  contin- 
ued product  of  the  divisors  into  the  numbers  in  the  last  tine, 
mil  be  the  least  common  multiple  required. 

16.  Find  the  least  common  multiple  of  6,  8,  and  12. 

First  Operation.  .  Second  Operation. 

2)6  "  8  "   12  6)6  /'  8  "   12 

2)3  "  4  "     6  2)1  "  8  /'     2 

3)3  "  2  '•     3  1  a  4  "     T 

*      1   "  2  "     1  Now  6x2x4=48. 

2x2x3x2=24  Ans. 

OBS.  1.  In  the  first  operation,  we  divide  by  the  smallest  num- 
bers which  will  divide  any  two  of  the  given  numbers  without  a  re- 
mainder, and  the  product  of  the  divisors,  and  the  numbers  in  the  last 
line,  is  24,  which  is  the  answer  required. 

In  the  second  operation,  we  divide  by  6,  then  by  2.  But  6  is  not 
the  smallest  number  that  will  exactly  divide  two  of  the  given  num- 
bers, and  the  continued  product  of  the  divisors  into  the  figures  in  the 
last  line  is  48,  which  is  not  the  least  common  multiple.  Hence, 

2.  We  must  divide,  in  all  cases,  by  the  smallest  number  that  will 
divide  any  two  of  the  given  numbers  exactly  ;  otherwise,  the  divisor 
may  contain  a  factor  common  to  it  and  some  one  of  the  quotients,  or 
undivided  numbers  in  the  last  line,  and  consequently  the  continued 
product  of  them  will  be  too  large  for  the  least  common  multiple.  Thus 
in  the  2d  operation,  the  6  and  4  contain  a  common  factor  2,  which 
must  be  rejected  from  them,  in  order  that  the  product  of  the  divisors 
Wid  quotients  may  be  the  least  common  multiple. 

17.  Find  the  least  common  multiple  of  4,  9,  and  12. 

18.  F'-id  the  least  common  multiple  of  16,  12,  and  24 

19.  Find  the  least  common  multiple  of  15,  9,  6,  and  5. 

20.  Find  the  least  common  multiple  of  10,  6,  18,  15. 

21.  Find  the  least  common  multiple  of  24,  16,  15,  20. 

22.  Find  the  least  common  multiple  of  25,  60,  72,  35. 

23.  Find  the  least  common  multiple  of  63,  12,  84,  72. 

24.  Find  the  least  common  multiple  of  54,  81,  14,  63. 

25.  Find  the  least  common  multiple  of  12,  72,  36,  144. 


QUEST. — Obs.  Why  do  you  divide  by  the  smallest  number  that  will 
iivide  two  or  more  without  a  remainder  ? 


100  FRACTIONS.  [SECT.  V] 

SECTION  VI. 
FRACTIONS. 

MENTAL    EXERCISES. 

ART.  1O3.  When  a  number  or  thing  is  divided  into 
two  equal  parts,  one  of  these  parts  is  called  one  half.  If 
the  number  or  thing  is  divided  into  three  equal  parts,  one 
of  the  parts  is  called  one  third ;  if  it  is  divided  into  four 
equal  parts,  one  of  the  parts  is  called  one  fourth,  or  one 
quarter;  two  of  the  parts,  two  fourths;  three,  three  fourths , 
if  divided  into  five  equal  parts,  the  parts  are  called  fifths  ; 
if  into  six  equal  parts,  sixths ;  if  into  ten,  tenths  ;  if  into  a 
hundred,  hundredths,  &c.  That  is, 

When  a  number  or  thing  is  divided  into  equal  parts,  the 
parts  always  take  their  name  from  the  number  of  parts  into 
which  the  thing  or  number  is  divided. 

1O4.  The  value  of  one  of  these  equal  parts  mani- 
festly depends  upon  the  number  of  parts  into  which  the 
given  number  or  thing  is  divided.  Thus,  if  an  orange  is 
successively  divided  into  2,  3,  4,  5,  6,  &c.,  equal  parts,  the 
thirds  will  be  less  than  the  halves  ;  the  fourths,  than  the 
thirds  ;  the  fifths,  than  the  fourths,  &c. 

Ex.   1.  What  is  one  half  of  2  cents  ?     Of  4  cents  ?  6 1 
8?   16?  18?  20?  24?  30?  40?  50?  60?  70?  80?  100? 
2.  What  is  one  third  of  6  cents  ?     Of  9?   12?   15? 


QUEST. — 103.  What  is  meant  by  one  half?  How  many  halves  make 
a  whole  one  ?  What  is  meant  by  one  third  ?  How  many  thirds  make 
a  whole  one  ?  What  is  meant  by  a  fourth  ?  3  fourths  ?  What  are 
fourths  sometimes  called?  How  many  fourths  make  a  whole  one? 
What  is  meant  by  fifths  ?  By  sixths  ?  Eighths  ?  How  many  sevenths 
make  a  whole  one  ?  How  many  tenths  ?  What  is  meant  by  twenti« 
eths  ?  By  hundredths  ?  When  a  number  or  thing  is  divided  into  equa. 
parts,  from  what  do  the  parts  take  their  name  ?  104.  Upon  what  does 
the  value  of  one  of  these  equal  parts  depend  ?  Which  is  the  greater,  a 
half  or  a  third  I  A  sixth  or  a  fourth  ?  A  seventh  or  a  tenth  1 


ARTS.  103,  104.]  FRACTIONS.  101 

OBS.  A  half  of  any  number,  it  will  be  perceived,  is  equal  to  as  many 
units  as  2  is  contained  times  in  that  number;  a  third  of  a  number  is 
equal  to  as  many  units,  as  3  is  contained  times  in  the  given  number; 
Q  fourth  is  equal  to  as  many,  as  4  is  contained  in  it,  &c. 

3.  What  is  a  third  of  12  ?    Of  15  ?  18  ?  21 1  24  ?  27  ? 
30?  36?  39?  45?  60? 

4.  What  is  a  fourth  of  8  dollars?     Of  12?  16?  20? 
24?  28?  32?  36?  40?  44?  48? 

5.  What  is  a  fifth  of  5  ?   10?   15?  20?  25?  30?  35? 
40?  45?  50?  55?  60?   100? 

6.  What  is  a  sixth  of  12?  18?  24?  36?  30?  48?  60? 
54?  42?  72? 

7.  What  is  a  seventh  of  14?  28?  35?  21?  42?  56? 
4$?  63? 

8.  What  is  an  eighth  of  16?  24?  40?  32?  64?  48? 
56?  72?  88? 

9.  What  is  a  ninth  of  9  ?   18?  36?  27?  45?  54?  72? 
63?  81?  99? 

10.  What  is  a  tenth  of  20  ?  40?  60?  50?  30?   100? 
90?  120? 

1 1 .  What  part  of  2  is  1  ?  Ans.  One  half. 

12.  What  part  of  3  is  1  ?  Of  4?  5?  7?   10?   15?  19? 
37?  200? 

13.  What  part  of  3  is  2? 

Suggestion. — Since  1  is  1  third  part  of  3,  2  must  be  two 
limes  the  third  part'of  3,  or  two  thirds  of  3. 

14.  What  part  of  5  is  2?  is  3?  is  4?  is  5?  is  6?  is 
8?  is9? 

15.  What  part  of  8  is  3  ?  is  7?  is  6?  is  9?  is  8?   12? 
? 

16.  What  part  of  17  is  5?  8?  9?   13?   15?   16?  20? 

17.  What  part  of  100  is  13?  29?  63?  75?  92? 

18.  If  1  half  an  orange  cost  2  cents,  what  will  a  whole 
orange  cost? 

Analysis. — If  1  half  of  an  orange  cost  2  cents,  2  halves 
or  a  whole  orange,  will  cost  twice  as  much ;  and  2  times 
2  cents  are  4  cents.  Ans.  4  cents. 


15? 


102  FRACTIONS.  [SECT.  VL 

19.  If  1  third  of  a  pie  cost  4  cents,  what  will  2  thirds 
cost  ?     What  will  a  whole  pie  cost  ? 

20.  If  1  fourth  of  a  pound  of  ginger  cost  3  cents,  whtt 
will  2  fourths  of  a  pound  cost  1  3  fourths  1     What  will  a 
whole  pound  cost  ? 

21.  If  1  eighth  of  a  yard  of  cloth  cost  2  shillings,  what 
will  3  eighths  cost?  5  eighths?  7  eighths'?   What  will  a 
whole  yard  cost  ? 

22.  If  1  third  of  a  barrel  of  flour  cost  3  dollars,  how 
much  will  a  whole  barrel  cost  ?     How  much  will  5  bar- 
rels cost  ?  8  barrels  1 

23.  If  1  sixth  of  a  hogshead  of  molasses  cost  5  dollars, 
what  will  be  the  cost  of  a  hogshead  ?     Of  4  hogsheads  ? 
Of  10  hogsheads? 

24.  If  1  pound  of  sugar  cost  12  cents,  what  will  1  half 
a  pound  cost  ? 

Suggestion. — If  1  pound  cost  12  cents,  it  is  plain  that 
1  half  of  a  pound  will  cost  1  half  of  12  cents  ;  and  1  half 
of  12  cents  is  6  cents.  Ans,  6  cents. 

25.  If  one  yard  of  ribbon  ^ost  15  cents,  how  much  will 

1  third  of  a  yard  cost  ? 

26.  If  one  pound  of  tea  cost  4  shillings,  how  much 
will  1  fourth  of  a  pound  cost  ?     How  much  will  2  fourths 
cost? 

27.  If  a  ton  of  hay  cost  15  dollars,  how  much  will  1 
fifth  of  a  ton  cost  ?     How  much  2  fifths  ?  3  fifths  ? 

28.  What  will  1  tenth  of  an  acre  of  land  cost,  at  30  dol 
lars  per  acre?  2  tenths ?  6  tenths  ? 

29.  What  will  1  eighth  of  a  ton  of  iron  cost,  at  48  dol- 
lars per  ton  ?  3  eighths  ?  5  eighths  ?  7  eighths  ? 

30.  If  1  bushel  of  corn  cost  1  half  a  dollar,  what  wiF 

2  bushels  cost  ?  4  bushels  ? 

Suggestion. — If  1  bushel  cost  1  half  a  dollar,  2  bushel* 
will  cost  twice  as  much.  2  times  1  half  are  2  halves,  or 
a  whole  dollar.  4  bushels  will  cost  4  times  1  half,  or  2 
whole  dollars. 

31.  If  one  man  eats  1  half  of  a  loaf  of  bread  at  a  meal, 
how  many  loaves  will  3  men  eat  ? 


ART.  104.]  FRACTIONS.  103 

32.  How  many  whole  ones  are  4  halves  equal  to  ?  5 
halves  ?  6  halves  ?  8  halves  1  9  halves  ? 

33.  If  I  burn  1  third  of  a  ton  of  coal  in  a  week,  how 
much  shall  I  burn  in  3  weeks  1  4  weeks  1  6  weeks  ?  10 
weeks?   12  weeks? 

34.  How  many  whole  ones  in  4  thirds,  and  how  many 
over?  In  6  thirds?  8  thirds?   11  thirds?   14  thirds? 

35.  If  a  horse  eat  1  fourth  of  a  bushel  of  oats  a  day, 
how  many  will  he  eat  in  6  days?  In  8  ?  In  10  ?   In  12? 

36.  If  a  boy  can  saw  1  eighth  of  a  cord  of  wood  in  a 
day,  how  much  can  he  saw  in  6  days?    In  12  days?  In 
15  days?   In  24  days? 

37.  If  12  oranges  were  divided  equally  among  4  boys, 
what  part  of  them  would  each  boy  receive;  and  how 
many  oranges  would  each  have  ? 

Analysis. — 1  is  1  fourth  of  4 ;  hence,  1  boy  must  re- 
ceive 1  fourth  part  of  the  oranges.  1  fourth  of  12  oran- 
ges is  3  oranges. 

38.  A  builder  employed  6  men  to  do  a  job  of  work,  for 
which  he  gave  them  24  dollars :  what  part  of  the  money 
did  1  man  receive  ?     What  part  did  2  receive  ?     What 
part  did  3  receive?     What  part  did  4  receive?     How 
many  dollars  did  one  man  receive  ?     How  many  did  two  ? 
Three  ?     Four  ? 

39.  If  5  yards  of  cloth  cost  40  dollars,  what  part  of 
40  dollars  will  1  yard  cost  ?  2  yards  ?  3  yards  ?  4  yards  ? 
How  many  dollars  will  1  yard  cost  ?  2  yards  ?  3  yards  ? 
4  yards  ? 

40.  2  is  1  third  of  what  number  ? 

Solution. — If  2  is  1  third  of  a  number,  3  thirds  or  the 
whole  number,  must  be  3  times  as  many. 

Or  thus,  2  is  a  third  of  3  times  2 ;  and  3  times  2  are  6. 

41.  4  is  1  fifth  of  what  number?   1  sixth  of  what  num 
oer  ?  1  third  ?   1  eighth  ?   1  fourth  ?  1  seventh  ? 

42.  6  is  1  third  of  what  number  ?  1  fourth  ?   1  seventh  ? 
I  tenth?   1  ninth?   1  twelfth? 

43.  5  is  1  fourth  of  what  number  ?  1  sixth  ?  1  eighth  ? 
eleventh  ?  1  twelfth  ? 


104  FRACTIONS.  [SECT.  VI, 

44.  8  is  1  seventh  of  what  number  ?   1  sixth  ?  1  tenth  ? 
1  ninth  ?  1  twelfth  ? 

45.  4  is  2  thirds  of  what  number  ? 

Suggestion. — First  find  1  third.  Now  if  4  is  2  thirds, 
1  third  is  1  half  of  4,  which  is  2 ;  and  3  thirds  is  3  times 
2,  or  6.  Ans.  6. 

46.  9  is  3  fourths  of  what  number  ? 

47.  8  is  4  fifths  of  what  number  1 

48.  16  is  4  ninths  of  what  number  ? 

49.  20  is  5  eighths  of  what  number  ? 

50.  32  is  8  twelfths  of  what  number  ? 

105.  When  a  number  or  thing  is  divided  into  equal 
parts,  as  halves,  thirds,  fourths,  &c.,  these  parts  are  called 
FRACTIONS. 

A  whole  number  is  called  an  Integer. 

106.  Fractions  are  divided  into  two  classes,  Com- 
mon and  Decimal.     (For  the  illustration  of  Decimal  Frac 
tions,  see  Section  VIII.) 

1O7«  Common  Fractions  are  expressed  by  two  num.' 
bers,  one  placed  over  the  other,  with  a  line  between  them. 
One  half  is  written  thus  \  ;  one  third,  i  ;  one  fourth,  \  , 
nine  tenths,  -^  ;  thirteen  forty-fifths,  if,  &c. 

The  number  below  the  line  is  called  the  denominator, 
and  shows  into  how  many  parts  the  number  or  thing  is 
divided. 

The  number  above  the  line  is  called  the  numerator,  and 
shows  how  many  parts  are  expressed  by  the  fraction, 
Thus  in  the  fraction  f ,  the  denominator  3,  shows  that  the 
number  is  divided  into  three  equal  parts ;  the  numerator 
2,  shows  that  two  of  those  parts  are  expressed  by  the 
the  fraction. 

The  denominator  and  numerator  together,  are  called 
the  terms  of  the  fraction. 


QUEST.— 105.  What  are  fractions  ?    What  is  an  integei       106.  Of 
cow  many  kinds  are  fractions  ?     107.  How  are  common  fractions  ex- 
pressed ?    What  is  the  number  below  the  line  called  ?    What  does  it 
show  ?   What  is  the  number  above  the  line  called  ?  What  does  it  show 
What  are  the  denominator  and  numerator,  taken  together,  called  ? 


ARTS.  105-110.]          FRACTIONS.  105 

OBS.  1.  The  term  fraction  is  of  a  Latin  origin,  and  signifies  brok- 
en, or  separated  into  parts.  Hence  fractions  are  sometimes  called 
broken  numbers. 

2.  Common  fractions  are  often  called  vulgar  fractions.     This  term, 
however,  is  very  properly  falling  into  disuse. 

3.  The  number  below  the  line  is  called  the  denominator,  because 
it  gives  the  name  or  denomination  to  the  fraction ;  as.  halves,  thirds, 
fifths,  &c. 

The  number  above  the  line  is  called  the  numerator,  because  it  num- 
bers the  parts,  or  shows  how  many  parts  are  expressed  by  the  fraction. 

108.  A  proper  fraction  is  a  fraction  whose  numer- 
ator is  less  than  its  denominator ;  as,  1,  f ,  |. 

An  improper  fraction  is  one  whose  numerator  is  equal 
to,  or  greater  than  its  denominator  ;  as,  f ,  |. 

A  mixed  number  is  a  whole  number  and  a  fraction  ex- 
pressed together  ;  as,  4|,  25-H-. 

A  simple  fraction  is  a  fraction  which  has  but  one  nu- 
merator and  one  denominator,  and  may  be  proper,  or 
improper ;  as,  f ,  -f-. 

A  compound  fraction  is  a  fraction  of  a  fraction  ;  as,  -£  of 
foff.  ^ 

A  complex  fraction  is  one  which  has  a  fraction  in  its 

2-1.  4  2^ 

numerator  or  denominator,  or  in  both ;  as,  — >  — »  — • 

5  0-3  84 

109.  Fractions,  it  will  be  seen,  both  from  the  defini- 
tion and  the  mode  of  expressing  them,  arise  from  division, 
and  may  be  treated    as  expressions  of  unexecuted  divis- 
ion, the  numerator  answering  to  the  dividend,  and  the 
denominator  to  the  divisor.  (Arts.  67,  105.)     Hence, 

110.  The  value  of   a  fraction  is  the  quotient  of  the 
numerator  divided  by  the  denominator.     Thus  the  value 
of  f  is  two ;  of  -f-  is  one ;  of -^  is  one  third ;  &c.     Hence, 


QUEST. — Obs.  What  is  the  meaning  of  the  term  fraction  ?  What 
Ere  common  fractions  sometimes  called  ?  Why  is  the  lower  number 
called  the.  denominator?  Why  is  the  upper  one  called  the  numerator  ? 
108.  What,  is  a  prope-  fraction?  An  improper  fraction?  A  mixed 
number?  A  simp'j  fraction?  A  con/pound  fraction?  A  complex 
fraction?  109.  From  what  do  fractions  arise  ?  110.  What  is  the  val- 
ue cf  a  fraction ! 


1 06  FRACTIONS.  [SECT.  VI. 

111.  If  the  denominator  remains  the  same,  multiplying 
he  numerator  by  any  number,  multiplies  the  value  of  the 
fraction  by  that  number.  For,  the  numerator  and  denom- 
inator answer  to  the  dividend  and  divisor ;  therefore, 
multiplying  the  numerator  is  the  same  as  multiplying  the 
dividend.  Now  multiplying  the  dividend,  we  have  seen, 
multiplies  the  quotient,  (Art.  83,)  which  is  the  same  as  the 
value  of  the  fraction.  (Art.  110.)  Thus,  the  value  of 
f =2.  Multiplying  the  numerator  by  3,  the  fraction  be- 
comes -^-,  whose  value  is  6,  and  is  the  same  as  2x3. 

112*  Dividing  the  numerator  by  any  number,  divide* 
the  value  of  the  fraction  by  that  number.  For,  dividing  the 
dividend  divides  the  quotient.  (Art.  84.)  Thus,  f=2. 
Now  dividing  the  numerator  by  2,  the  fraction  becomes 
•f,  whose  value  is  1,  and  is  the  same  as  2-^-2.  Hence, 

OBS.    With  a  given  denominator,  the  greater  the  numerator,  the 
greater  will  be  the  value  of  the  fraction. 

113*  If  the  numerator  remains  the  same,  multiplying 
the  denominator  by  any  number,  divides  the  value  of  the 
fraction  by  that  number.  For,  multiplying  the  divisor 
divides  the  quotient.  (Art.  85.)  Thus,  •^•=4.  Now 
multiplying  the  denominator  by  2,  the  fraction  becomes 
•f£,  whose  value  is  2,  and  is  the  same  as  4-J-2. 

114.  Dividing  the  denominator  by  any  number,  mul- 
tiplies the  'value  of  the  fraction  by  that  number.  For,  divi- 
ding the  divisor,  multiplies  the  quotient.  (Art.  86.) 
Thus,  •^L=4.  Now  dividing  the  denominator  by  2,  the 
fraction  becomes  -^  whose  value  is  8,  and  is  the  same  as 
4x2.  Hence, 

OBS.  With  a  given  numerator,  the  greater  the  denominator,  the 
less  will  be  the  value  of  the  fraction. 


QUKST. — 111.  What  is  the  effect  of  multiplying  the  numerator,  while 
the  denominator  remains  the  same  1  Explain  the  reason.  112.  What 
is  the  effect  of  dividing  the  numerator  \  Obs.  With  a  given  denomin- 
ator, what  is  the  effect  of  increasing  the  numerator?  113.  What  is 
the  effect  of  multiplying  the  denominator  ?  Why  I  1 14.  What  is  the 
effect  of  dividing  the  denominator  ?  Why?  Obs.  With  a  given  nu> 
merator,  what  is  tho  effect  of  increasing  the  denominator  ? 


ARTS.  111-118.]          FRACTIONS.  107 

115.  It  is  evident  from  the  preceding  articles,  that 
multiplying  the  numerator  by  any  number,  has  the  same 
effect  on  the  value  of  the  fraction,  as  dividing  the  denomi- 
nator  by  that  number.    (Arts.  Ill,  114.) 

Dividing  the  numerator  has  the  .same  effect,  as  multi 
plying  the  denominator.   (Arts.  112,  Il3.) 

116.  If  the  numerator  and  denominator  are  both 
multiplied  or  both  divided  by  the  same  number,  the  value 
of  the  fraction  will  not  be  altered.    (Arts.  88,  109.)    Thus, 
•^=3.     Now  if  the  numerator  and  denominator  are  both 
multiplied  by  2,  the  fraction  becomes  *£•  ;  whose  value  is 
3.     If  both  terms  are  divided  by  2,  the  fraction  becomes 
f  ,  whose  value  is  3  ;  that  is,  Y=^-=f  =3. 


117.  Since  the  value  of  a  fraction  is  the  quotient  of 
the  numerator  divided  by  the  denominator,  it  follows,  that 

If  the  numerator  and  denominator  are  equal,  the  value 
is  a  unit  or  one.  Thus,  f=l,  -f=l,  &c. 

If  the  numerator  is  greater  than  the  denominator,  the 
value  is  greater  than  one.  Thus,  f  =2,  f  =l-f. 

If  the  numerator  is  less  than  the  denominator,  the 
value  is  less  than  one.  Thus,  ^-=1  third  of  1,  -£=4  fifths 
of  1. 

118.  It  will  be  seen  from  the  preceding  exercises, 
that  fractions  may  be  added,  subtracted,  multiplied,  and  di- 
vided, as  well  as  whole  numbers. 

OBS.  1.  In  order  to  perform  these  operations,  it  is  often  necessary 
to  make  certain  changes  in  the  terms  of  the  fractions. 


QUEST. — 115.  What  maybe  done  to  the  denominator  to  produce  the 
earne  effect  on  the  value  of  the  fraction,  as  multiplying  the  numerator 
by  any  given  number  ?  What,  to  produce  the  same  effect  as  dividing 
the  numerator  by  any  given  number  ?  1 16.  What  is  the  effect  if  the 
numerator  and  denominator  are  both  multiplied,  or  both  divided  by  the 
same  number  ?  117.  When  the  numerator  and  denominator  are  equal, 
what  is  the  value  of  the  fraction  ?  When  the  numerator  is  the  larger, 
ivhat  ?  When  smaller,  what  1 


108  REDUCTION    OF  [SECT.  VI, 

2.  It  is  evident  that  any  changes  may  be  made  in  the  terms  of  « 
fraction,  which  do  not  alter  the  quotient  of  the  numerator  divideq 
by  the  denominator;  for,  if  the  quotient  is  not  altered,  the  val- 
ue remains  the  same.  (Art.  1  10.^  Thus,  the  terms  of  the  fraction 
^  may  be  changed  into  -2,  -S.,  JUI,  &c.,  without  altering  its  value  ;  for 
in  each  case  the  quotient  of  the  numerator  divided  by  the  denomin- 
ator is  2.  Hence,  for  any  given  fraction,  we  may  substitute  any 
other  fraction,  which  will  give  the  same  quotient. 

REDUCTION  OF  FRACTIONS. 

119*  The  process  of  changing  the  terms  of  a  fraction 
into  others,  without  altering  its  value,  is  called  REDUCTION 
OF  FRACTIONS. 

EXERCISES  FOR  THE  SLATE. 

CASE     I. 

Ex.  1.  Reduce  ~fa  to  its  lowest  terms. 

Dividing1  both  terms  of 
the  fraction   by   2,  it  be- 
=f:  again,  3)f  =£.  Ans.     comes  A.   againj  dividing 

both  by  3,  we  obtain  £,  whose  terms  are  the  lowest  tc 
which  the  given  fraction  can  be  reduced. 

If  we  divide  both  terms  by  6,  their 
Second  Operation. 


common 

—  \-  Ans.        the  given  fraction  will  be  reduced  to 
its  lowest  terms  by  a  single  division.     Hence, 

1  2O.  To  reduce  a  fraction  to  its  lowest  terms. 

Divide  the  numerator  and  denominator  by  any  number 
which  will  divide  them  both  without  a  remainder  ;  and  thus 
continue  the  operation,  till  there  is  no  number  greater  than  1 
that  will  divide  tJiem  exactly. 

Or,  divide  both  the  numerator  and  denominator  by  thei? 
greatest  common  divisor  ;  and  the  two  quotients  thus  atising 
will  be  the  lowest  terms  to  which  the  given  fraction  can  be  re 
duced.  (Art.  96.) 


QUEST. — Obs.  What  changes  may  be  made  in  the  terms  of  a  frac 
lion  1  119.  What  is  meant  by  reduction  of  fractions?  120.  How  is  q 
fraction  reduced  to  its  lowest  terms  ? 


ARTS.  119-121.  ]         FRACTIONS.  109 

OBS.  1.  A  fraction  is  said  to  be  reduced  to  its  lowest  terms,  when 
Hs  numerator  and  denominator  are  expressed  in  tne  smallest  num- 
bers possible. 

2.  The  value  of  a  fraction  is  not  altered  by  reducing  it  to  its  lowest 
terms.  (Art.  116.) 

3.  Wnen  the  terms  of  the  fraction  are  small,  the  former  method  will 
generally  be  found  to  be  the  shorter  and  more  convenient;  but  when 
the  terms  are  large,  it  is  often  difficult  to  determine  whether  the  frac- 
tion is  in  its  simplest  form,  without  finding  their  greatest  common 
divisor. 

2.  Reduce  -ft-  to  its  lowest  terms.  A?is.  £. 

3.  Reduce  -ft-.  4.  Reduce  f  . 
5.  Reduce  -'if.  6.  Reduce  ft. 
7.  Reduce  if.  8.  Reduce  £f 
9.  Reduce  -ftfe.                   10.  Reduce 

11.  Reduce  -fff-  12.  Reduce 

13.  Reduce  f££  14.  Reduce 

15.  Reduce-]3^.  16-  Reduce 

17.  Reduce  -.«.  18.  Reduce 


CASE   II. 
19.  Reduce  -Y1  to  a  whole  or  mixed  number. 

Suggestion.  —  The  object  in  this  example,  is  to  Operation. 
find  a  whole  or  mixed  number,  whose  value  is 

equal  to  the  given  fraction.     But  the  value  of  a  5)17 

fraction  is  the  quotient  of  the  numerator  divided  ~  q  a 

by  the  denominator.     (Art.  110.)     Hence,  6* 


To  reduce  an  improper  fraction  to  a  whole,  or 
mixed  number. 

Dityide  the  numerator  by  the  denominator,  and  &•  '^ 
iient  will  be  the  whole,  or  mixed  number  required. 

20.  Reduce  -^  to  a  whole  or  mixed  number. 

Ans.  9|. 


QUEST. —  Obs.  What  is  meant  by  lowest  terms  t  Is  the  value  of 
A  fraction  altered  by  reducing  it  to  its  lowest  terms  ?  121.  How  if 
an  improper  fraction  reduced  to  a  whole  or  mixed  number  ? 


HO  REDUCTION  OF  [SECT.  VI. 

Reduce  the  following  fractions  to  whole  or  mixed 
numbers : 

21.  Reduce  *£•.  22.  Reduce  ty. 

23.  Reduce  *£-.  24.  Reduce  ±A. 

25.  Reduce  -H-.  26.  Reduce 

27.  Reduce  ^.  28.  Reduce 

29.  Reduce  *&.  30.  Reduce 

CASE   III. 

31.  Reduce  the  mixed  number  15f  to  an  improper 
fraction. 

Operation. 

15_a  OBS.  In  1  there  are  4  fourths,  and  in  15,  there  are 

15  times  as  many.    4X15=60,  and  3  fourths  make 
__!  63  fourths.     Hence, 

^  Ans. 

122*  To  reduce  a  mixed  number  to  an  improper 
fraction. 

Multiply  the  whole  number  by  the  denominator  of  the  frac- 
tion ;  to  the  product  add  the  given  numerator.  The  sum 
placed  over  the  given  denominator,  ivill  form  the  improper 
fraction  required. 

OBS.  1.  Any  whole  number  may  be  expressed  in  the  form  of  a  frac- 
tion without  altering  its  value,  by  making  \  the  denominator. 

2.  A  whole  number  may  also"  be  reduced  to  a  fraction  of  any  de- 
nomination, by  multiplying  the  given  number  by  the  proposed  denom- 
inator; the  product  will  be  the  numerator  of  the  fraction  required. 

Thus  25  may  be  expressed  by  -*£-,  -"-f0-,  or  *£?-,  &c.,  foi 
25=^^=4^=4^1,  &c.  So  12=Ve=¥=¥=¥-;  fo^  the 
quotient  of  each  of  these  numerators  divided  by  its  de- 
nominator, is  12. 

32.  Reduce  8-£  to  an  improper  fraction.         Ans.  ^-. 


QUEST. — 122.  How  reduce  a  mixed  number  to  an  improper  rraction  ? 
Obs.  How  express  a  whole  number  in  the  form  of  a  fraction  I  How  re- 
duce a  whole  number  to  a  fraction  of  a  given  denominator  ? 


AETS.  122,  123.]         FRACTIONS.  Ill 

Reduce  the  following  numbers  to  improper  fractions: 

33.  Reduce  9-f.  34.  Reduce  16*. 

35.  Reduce  23f  36.  Reduce  45^. 

37.  Reduce  64^.  38.  Reduce  56ff. 

39.  Reduce  304^  40.  Reduce  725£. 

.    41.  Reduce  45  to  fifths.  42.  Reduce  72  to  eighths. 

43.  Reduce  830  to  sixths. 

44.  Reduce  743  to  fifteenths 

CASE   IV. 

45.  Reduce  f  of  f  to  a  simple  fraction. 

Analysis. — f  of  f  is  2  times  as  much  as  1  third  of  -f-. 
Now  i  of  -f-  is  -fa  ]  for,  multiplying  the  denominator  di 
vides  the  value  of  the  fraction.  (Art.  1 13.)  And  2  thirds 
is  2  times  -/v,  which  is  equal  to  it,  or  f  (Art.  120.) 
The  answer  is  ^. 

OBS.  This  operation  consists  in  simply  multiplying  the  two  nu 
•iterators  together  and  the  two  denominators.  Hence, 

123*  To  reduce  compound  fractions  to  simple  ones. 

Multiply  all  the  numerators  together  for  a  new  numera- 
tor, arid  all  the  denominators  together  for  a  new  denomv 
nator. 

46.  Reduce  -f  of  f  of  -f  to  a  simple  fraction. 

Ans.  jVV,  or  ^. 

47.  Reduce  |  of  if  of  W  to  a  simple  frnction. 

48.  Reduce  •§•  of  f  of  $  to  a  simple  fraction. 

49.  Reduce  f  of  -^  of  -fi  to  a  simple  fraction. 

50.  Reduce  •£  of  £  of  f  of  -f-  to  a  simple  fraction. 
Operation.  Since  the  product  of  the  numera- 

^  A  *  5  5  tors  is  to  be  divided  by  the  product 
-  of -of  -of-=  -  of  the  denominators,  we  may  can- 
£  $  4  7  28  ce[  tjie  factors  2  and  3,  which  are 
common  to  both ;  for,  this  is  dividing  the  terms  of  the  new 
fraction  by  the  same  number,  (Art.  90,)  and  therefore 
aloes  not  alter  its  value.  (Art.  116.)  Multiplying  the  re- 

QUEST. — 123.  How  are  compound  fractions  reduced  to  simple  ones  1 


113  BEDUCTIOW  OP  [SECT.  VL 

maining1  factors  together,  we  have  -fa,  which  is  the  ai> 
swer  required.  Hence, 

124*  To  reduce  compound  fractions  to  simple  ones 
by  CANCELATION. 

Cancel  all  the  factors  which  are  common  to  the  numer- 
ators and  denominators  ;  then  multiply  the  remaining  terms 
together  as  before.  (Art.  123.) 

OBS.  This  method  not  only  shortens  the  operation  of  multiplying 
but  at  the  same  time  reduces  the  answer  to  its  lowest  terms.  A  little 
practice  will  give  the  learner  great  facility  in  its  application. 

51.  Reduce  i  of  -ff  of  f  to  a  simple  fraction. 

Operation.  First,  we  cancel  the  4  and  3 

2  in  the  numerator,  then  the  12  in 

^      £0       &     2  t^ie  denominator,  which  is  equal 

-rof  -^of  -;===  A™.        to  the  factors  4  and  3.     Final- 

ly, we  cancel  the  5  in  the  de- 

nominator, and  the  factor  5  in  the  numerator  10,  placing 
the  other  factor  2  above.  We  have  2  left  in  the  numera- 
tor and  7  in  the  denominator.  Ans.  -f 


52.  Reduce  f  off  of  if  to  a  simple  fraction. 

53.  Reduce  $  of  -J-  of  -fa  of  •£  to  a  simple  fraction. 

54.  Reduce  -f  of  -f-  of  •§•  of  -^  to  a  simple  fraction. 

55.  Reduce  -ft-  of  •$•£  of  -ff  to  a  simple  fraction. 

56.  Reduce  -fa  of  -ft  of  -f-  of  ^  to  a  simple  fraction 

57.  Reduce  -f  of  -ff-  of  -ff  of  -^  to  a  simple  fraction. 

58.  Reduce  -fa  of  -fa  of  -f-  of  -f  to  a  simple  fraction. 

Note.  —  For  the  method  of  reducing  complex  fractions  to  simple  onea 
see  Art.  143. 

CASE   V. 
Ex.  1.  Reduce  £  and  •£•  to  a  common  denominator. 

Note.  —  Two  or  more  fractions  are  said  to  have  a  common,  denon* 
inatw,  when  they  have  the  same  denominator. 

QUEST  —  124.  How  by  cancelation  ?  How  does  it  appear  that  tlu# 
method  does  not  alter  the  value  of  the  fraction  ?  Obs.  What  is  tha 
advantage  of  tliis  method  ?  Note.  What  is  meant  by  a  common  de- 
nominator ? 


ART.  124,  125.]  FRACTIONS.  113 

Suggestion,. — The  object  of  this  example  is  to  find  two 
ether  fractions,  which  have  the  same  denominator,  and 
whose  values  are  respectively  equal  to  the  values  of  the 
given  fractions,  £  and  •£.  Now,  if  both  terms  of  the  first 
fraction  -£-,  are  multiplied  by  the  denominator  of  the  sec- 
ond, it  becomes  £•,  and  if  both  terms  of  the  second  fraction 
•£-,  are  multiplied  by  the  denominator  of  the  first,  it  be- 
comes -f.  But  the  fractions  f-  and  -f  have  a  common  de- 
nominator, and  are  respectively  equal  to  the  given  fractions, 
viz:  f=£,  andf=i.  (Art.  116.)  Hence, 

125.  To  reduce  fractions  to  a  common  denominator. 

Multiply  each  numerator  into  all  the  denominators  except 
its  own  for  a  new  numerator,  and  all  the  denominators  together 
for  a  common  denominator. 

2.  Reduce  •£,  -f- ,  and  -J-  to  a  common  denominator. 

Operation. 

I  x4x6=24  } 

3x2x6=36  >  the  three  numerators. 

5x2x4=40  ) 

2x4x6=48,  the  common  denominator. 

The  fractions  required  are  -f^-,  -f-f-,  and  •£§. 

OBS.  It  is  manifest  that  the  process  of  reducing  fractions  to  a  com- 
mon denominator,  does  not  change  their  value ;  for,  it  is  simply  mijl- 
tiplying  each  numerator  and  denominator  of  the  given  fractions  by  the 
game  number.  (Art.  116.) 

3.  Reduce  -f,  f ,  and  -f-  to  a  common  denominator. 

Ans.  -Mr,  tW,  -tffr. 

4.  Reduce  -f ,  -f,  and  f  to  a  common  denominator. 
Reduce  the  following  fractions  to  a  common  denomi- 
nator : 

5.  Reduce  |,  i  £,  and  f .      6.  Reduce  -f,  -*-,  f,  and  f. 
7.  Reduce -£,f,^V,  and -A.  *.  Reduce -ft, f, if,  and*. 


QUEST. — 125.  How  are  fractions  reduced  to  a  common  denominator  ? 
06s.  Does  the  process  of  reducing  fractions  to  a  common  denominator 
alter  their  value  ?  Why  not  ? 


114  REDUCTION   OP  [SECT.    VI 

9.  Reduce  £f,  fjf-,  and  ££    10.  Reduce  •&,  -fifr,  and  ff 
1 1.  Reduce  /r,  -f-fr,  and  if.    12.  Reduce  ^V,  •§*  and  iff. 


CASE   VI. 

13.  Reduce  f,  -£,  and  •£  to  the  least  common  denomi- 
nator. 

Operation.  We  first  find  the  least  common 

2)4  "  6  "  8  multiple  of  all  the  given  denomi- 

2)2  "  3  "  4  nators,  which  is  24;  (Art.  102;) 

0  7  and  this  is  the  least  common  de- 

nominator  required.     The   next 
24,  the  least          ig  tQ  re(£ce  the     iyen  frac 

tions  to  twenty-fourths  without  al- 

tering  their  value.  This  may  evidently  be  done,  by  mul- 
tiplying both  terms  of  each  fraction  by  the  number  of  times 
its  denominator  is  contained  in  24.  (Art.  116.)  Thus  4, 
the  denominator  of  the  first  fraction,  is  contained  in  24,  6 
times ;  now  multiplying  both  terms  of  the  fraction  -f  by  6, 
it  becomes  -^f.  The  denominator  6  is  contained  in  24,  4 
times  ;  and  multiplying  the  second  fraction  -f-  by  4,  it  be- 
comes -^4-.  The  denominator  8  is  contained  in  24, 3  times ; 
and  multiplying  the  third  fraction  •£  by  3,  it  becomes  -J-f-. 
Therefore  -J-f,  -tfa,  and  -J-f-  are  the  fractions  required.  Hence. 

'   126.  To  reduce  fractions  to  their  least  common  de- 
nominator. 

I.  Find  the  least  common  multiple  of  all  the  denominator? 
of  the  given  fractions,  and  it  will  be  the  least  common  denomi- 
nator.    (Art.  102.) 

II.  Divide  the  least  common  denominator  by  the  denomi- 
nator of  each  of  the  given  fractions,  and  multiply  the  quotient 
by  the  numerator ; — the  products  icill  be  the  numerators  re- 
quired. 


QUEST-     126.  How  ?ire  fractions  reduced  to  the  least  common  de 
nominator  ? 


ART.  126.]  FRACTIONS.  115 

OBS.  Multiplying  each  numerator  into  the  number  of  tunes  its  de- 
nominator is  contained  in  the  least  common  denominator,  is  in  effect 
multiplying  both  terms  of  the  given  fractions  by  the  same  number. 
For,  if  we  multiply  each  denominator  by  the  number  of  times  it  is 
contained  in  the  least  common  denominator,  the  product  will  be  equal 
to  the  least  common  denominator.  Hence,  the  new  fractions  must  be 
of  the  same  value  as  the  given  fractions.  (Art.  116.) 

14.  Reduce  f,  f,  and  £  to  the  least  com.  denominator. 

Operation,  2x3x2=12,  the  least  com.  denominator. 

2^3  "  4  "  6  ^T°w  (12-f-3)x2=8,  numerator  of  1st. 
o»  „  o  ••  3  (12-r4)X3=9,  of2d. 

3)3      *      6  (12-T-6)X5=10,        «         of  3d. 

1  "  2  "  1  Ans.  A,  A,  and  if. 

15.  Reduce  •£  and  -ft  to  the  least  common  denominator. 

Ans.  -fi  and  -f$. 

Reduce  the  following  fractions  to  the  least  common  de- 
nominator : 

16.  i,  £ ,  and  f  17.  -f,  f ,  and  -ft. 

1 8.  f ,  -I,  A,  and  Tft-.  19.  *,  f ,  -ft,  and  A- 

20.  A,  i,  *,  i,  and  f  21.  -^,  ^-,  and  &. 

22.  Tft,  A,  and  -rib.  23.  -Ji,  f ,  and  -ft. 

24.  f ,  A,  and  if.  25.  -ft,  A,  and  ffr. 


ADDITION  OF  FRACTIONS. 
MENTAL    EXERCISES. 

Ex.  1.  What  is  the  sum  of  i,  •£,  f,  and  -f? 

Suggestion. — Since  all  these  fractions  have  the  same  de- 
nominator, it  is  plain  their  numerators  may  be  added  as 
well  as  so  many  pounds  or  bushels,  and  their  sum  placed 
over  the  common  denominator,  will  be  the  answer  re- 
quired. Thus,  1  eighth  and  2  eighths  are  3  eighths,  and 
3  are  6  eighths,  and  5  are  1 1  eighths.  Ans.  -V",  or  1-f . 


QUEST. — Obs.  Does  this  process  alter  the  value  of  the  given  frac- 
tions?   Why  not? 


116  ADDITION   OF  [SECT.   VL 

2.  What  is  the  sum  of  i,  -f-,  f ,  and  f  ? 

3.  What  is  the  sum  of  f ,  f ,  •£-,  and  -f  ? 

4.  What  is  the  sum  of  -ft,  ii,  -ft,  and 

5.  What  is  the  sum  of  -ft-,  -ft-,  -ft-,  -ft,  and 

6.  What  is  the  sum  of  -&,  •&,  ^,  if,  and 

7.  What  is  the  sum  of  if,  -ft,  Jf,  -ft-,  and 

8.  What  is  the  sum  of  ££,  fj-,  ^  A,  and  -6\  ? 

9.  What  is  the  sum  of  if,  if,  ^y,  A,  and  •&  ? 

10.  What  is  the  sum  of -fifty,  ifo,  iW,  and  -rfjr  ? 

EXERCISES   FOR   THE   SLATE. 

1 1.  What  is  the  sum  of  -f,  £,  and  f  ? 
Solution. — f+i+f =f,  or  If  ^ws. 

12.  What  is  the  sum  of  •£,  and  i  ? 

Suggestion. — A  difficulty  here  presents  itself;  for  it  13 
manifest  that  1  half  added  to  1  third  will  make  neither  2 
halves  nor  2  thirds.  (Art.  22.)  This  difficulty  may  be 
removed  by  reducing  the  given  fractions  to  a  common  da 
nominator.  (Art.  125.)  Thus, 

1x3=3  } 

1x2=2  \  ^e  new  numerators- 

2x3=6,  the  common  denominator. 

The  fractions  reduced  are  -f-  and  •£.  and  may  now  b<» 
added.  Thus,  f -ff =£ .  Ans. 

127*  From  these  illustrations  we  deduce  the  follow- 
ing general 

RULE  FOR  ADDITION  OF  FRACTIONS. 

Reduce  the  fractions  to  a  common  denominator  ;  &dd  their 
numerators,  and  place  the  sum  over  the  common  denomi- 
nator. 

OBS.  1.  Compound  fractions  must,  of  course,  be  reduced  to  simple 
ones,  before  attempting  to  reduce  the  given  fractions  to  a  common  de« 
nominator.  (Art.  123.) 

QUEST.— 127.  How  are  fractions  added  ?  Obs.  What  must  be  dona 
with  compound  fractions  ? 


ART.  127.]  FRACTIONS.  117 

2.  Mixed  numbert  may  be  reduced  to  improper  fractions,  then  added 
according  to  the  rule ;  or,  we  may  add  the  whole  numbers  and  frae« 
tional  parts  separately,  and  then  unite  their  sums. 

13.  What  is  the  sum  off,  and  f  ?      Ans.  £=!£ ,  or  1£, 

14  What  is  the  sum  o£  f,  and  •£  ? 

15.  What  is  the  sum  off,  •£,  and  -f  ? 

16.  What  is  the  sum  off  -H,  and  £  ? 

17.  What  is  the  sum  of  -&,  f,  and  -^  ? 

18.  What  is  the  sum  of  f ,  -ft,  and  T^  ? 

19.  What  is  the  sum  of  ^f,  f ,  and  f  ? 

20.  What  is  the  sum  of  ^  -f,  and  -^  ? 

21.  What  is  the  sum  of  f ,  f,  -f,  and  -f  ? 

22.  What  is  the  sum  of  -J-,  f ,  f ,  and  -f-  ? 

23.  What  is  the  sum  of  f ,  -f  of  £,  and  -&  ? 

24.  What  is  the  sum  of  f ,  -f ,  -f  of  f ,  and  f  ? 

25.  What  is  the  sum  of  i  of  3,  f  of  f,  and  f  ? 

26.  What  is  the  sum  of  2£,  6i,  and  f  ? 

27.  What  is  the  sum  of  f  of  2,  3£,  and  5f  ? 

28.  What  is  the  sum  of  If,  ff ,  and  -^  ? 

29.  What  is  the  sum  of  351,  -fi,  and  -f  of  f  ? 

30.  What  is  the  sum  of  -^-,  &J-.  If,  and  f  ? 

SUBTRACTION  OF  FRACTIONS. 

MENTAL   EXERCISES. 

Ex.  1.  Henry  had  -f-  of  a  watermelon,  and  gave  away 
£  of  it :  how  much  had  he  left  ? 

Solution. — 3  sevenths  from  5  sevenths  leaves  2  sevenths. 

Ans.  f 

2.  John  had  -£  of  a  bushel  of  chestnuts,  and  gave  away 
f- :  how  many  had  he  left  ? 

3.  If  I  own  f  of  an  acre  of  land,  and  sell  f  of  it,  how 
much  shall  I  have  left  1 

QUEST. — Obs.  How  are  mixed  numbers  added  ? 


118  SUBTRACTION   OF  [SECT.  YL 

4.  A  man  owning  •£  of  a  ship,  sold  -f  :  what  part  of 
the  ship  had  he  left  ? 

5.  William  had  -ft  of  a  dollar,  and  spent  -fa  :  how 
many  tenths  had  he  left  ? 

6.  What  is  the  difference  between  -fc  and 

7.  What  is  the  difference  between  £§•  and 

8.  What  is  the  difference  between  -JHr  and  ft  ? 

9.  What  is  the  difference  between  ii  and  fg-  ? 
10.  What  is  the  difference  between  T3^  and 


EXERCISES   FOR    THE    SLATE. 

11.  From-j^-  take  -fa. 
Solution.  —  $3  —  &*=-$[•  Ans. 

12.  From  -f-  take  f. 

Suggestion.  —  A  difficulty  here  meets  the  learner,  simi 
lar  to  that  which  occurred  in  the  12th  example  of  addi- 
tion of  fractions,  viz  :  that  of  subtracting  a  fraction  of 
one  denominator  from  a  fraction  of  a  different  denomina^ 
tor.  He  must  therefore  reduce  the  fractions  to  a  com 
mon  denominator,  before  the  subtraction  can  be  per 
formed. 


Also  6x4=24,  the  common  denominator. 
The  fractions  are  f£  and  if.      Now  -^  —  &=&•  Ans 


12S.  From  these  illustrations  we  deduce  the  follow 
ing  general 

RULE  FOR  SUBTRACTION  OF  FRACTIONS. 

Reduce  the  given  fractions  to  a  common  denominator  ;  sub- 
tract the  less  numerator  from  the  greater,  and  place  the  remain- 
der over  the  common  denominator. 

OBS.  Compound  fractions  must  be  reduced  to  simple  ones,  as  in  ad- 
dition of  fractions.  (Art.  123.) 

QUEST.  —  128.  How  is  one  fraction  subtracted  from  another  ?  O5* 
What  Is  to  be  done  with  compound  fractions  \ 


ARTS.  128,  129.]          FRACTIONS.  119 

13    From  -f  talre  i.  Ans.  £. 

14.  From  f  take  f.  15.  From  -f£  take  -ft. 

16.  From  it  take  f.  17.  From  if  take  \. 

18.  From  $f  take  -&&.  19.  From  if  take  •&. 

20.  From  &  take  ^  21.  From  f  take  -ft. 

22.  From  ff  take  ff.  23.  From  -ff-  take  -&. 

129.  Mixed  numbers  may  be  reduced  to  improper 
fractions,  then  to  a  common  denominator  and  subtracted  ; 
or,  the  fractional  part  of  the  less  number  may  be  taken 
from  the  fractional  part  of  the  greater,  a^d  the  less  whole 
number  from  the  greater. 

24.  From  8$-  take  5|. 
Operation. 

17  thirds  from   25   thirds   leaves   8 
thirds,  which  are  equal  to  2f. 


A    f»»        QJL  Note.  —  Since  we  cannot  take  2  thirds  from  1 

>r  tims,  o-  thir(J)  we  borrow  a  unitj  which,  reduced  to  thirds 

5f  and  added  to  1  third,  makes  4  thirds.    Now  2 

j-wc  Oi  thirds  from  4  thirds  leaves  2  thirds  :  1  to  carry  to 

4  8  5  makes  6,  and  6  from  8  leaves  2. 

25.  From  12-f  take  7-J-.     Ans.  5-f. 
!26.  From  1  5-f-  take  9£. 

27.  From  25f  take  17f. 

28.  From  37-$-  take  19-f-. 

29.  From  2  take  f  . 

Suggestion.  —  Since  5  fifths  make  a  whole  one,  in  2 
whole  ones  there  are  10  fifths  ;  now  3  fifths  from  10  fifths 
leaves  7  fifths.  Ans.  •£,  or  If.  Hence, 


QUEST.— 129.  How  are  mixed  numbers  subtracted  ?     130.  How  is  a 
fraction  subtracted  from  a  whole  number  ? 


120  MULTIPLICATION    OF  [SECT.  VI, 

13O*  To  subtract  a  fraction  from  a  whole  number. 

Change  the  whole  number  to  a  fraction  having  the  samt 
denominator  as  the  fraction  to  be  subtracted,  and  proceed  a\ 
before.  (Art.  128.) 

OBS.  If  the  fraction  to  be  subtracted  is  a  proper  fraction,  we  may 
simply  borrow  a  unit  and  take  the  fraction  from  this,  remembering  to 
diminish  the  whole  number  by  1.  (Art.  36.) 

30.  From  6  take  -f.  Ans.  5£ 

31.  From  65  take  25 W. 

32.  From-f  off  take  i  off. 

33.  From  i  off  take  i  of  T\. 

34.  From  f  of  10  take  f  of  6. 

35.  From  f  of  24  take  f  of  27. 

MULTIPLICATION  OF  FRACTIONS. 

MENTAL    EXERCISES. 

1.  It  a  man  spends  •£  of  a  dollar  for  rum  in  1  day,  how 
much  will  he  spend  in  7  days  ? 

Suggestion. — If  he  spends  i  in  1  day,  in  7  days  he  will 
spend  7  times  i  ;  and  ix7  is  •£.  Ans.  i  of  a  dollar. 

2.  If  a  man  spends  •£  of  a  dollar  for  rum  in  1  week, 
how  much  will  he  spe-nd  in  4  weeks.     Ans.  •*£•  or  3£  dolls. 

3.  If  1  man  drinks  -f  of  a  barrel  of  beer  in  a  month, 
how  much  will  10  men  drink  in  the  same  time  ? 

4.  What  cost  4  yards  of  cloth,  at  2£  dollars  per  yard  ? 

Solution. — 4  yards  will  cost  4  times  as  much  as  1  yard  ; 
and  4  times  i  is  4  halves,  equal  to  two  whole  ones :  4 
times  2  dollars  are  8  dollars,  and  2  i?>*?Ve.  10  dollars. 

Ans.  4  yards  will  cost  10  aoiiars. 

5.  What  cost  5i  bushels  of  peanuts,  at  3  dolls,  a  bushel  ? 

6.  What  cost  10-f  pounds  of  tea,  at  4  shillings  a  pound  1 

7.  If  1  drum  of  figs  costs  16  shillings,  what  will  3 
fourths  of  a  drum  cost  ? 

Suggestion. — First  find  what  1  fourth  will  cost.  Then 
3  fourths  will  cost  3  times  as  much. 


ARTS.  130-132.]          FRACTIONS.  121 

8.  If  an  acre  of  land  produces  40  bushels  of  corn,  how 
many  bushels  will  3  eighths  of  an  acre  produce  ? 

9.  If  a  man  can  travel  50  miles  in  a  day,  how  far  can 
he  travel  in  2  fifths  of  a  day?  3  fifths?  4  fifths? 

10.  Henry's  kite  line  was  90  feet  long,  but  getting  en- 
tangled in  a  tree,  he  lost  3  ninths  of  it :  how  many  feet 
did  he  lose  ? 

131.  We  have  seen  that  multiplying  by  a  whole 
number  is  taking  the  multiplicand  as  many  times  as  there 
are  units  in  the  multiplier.  (Art.  45.)  On  the  other 
hand, 

If  the  multiplier  is  only  a  part  of  a  unit,  it  is  plain  we 
must  take  only  a  part  of  the  multiplicand.  That  is, 

132*  Multiplying  by  a  fraction  is  taking  a  certain 
PORTION  of  the  multiplicand  as  many  times  as  there  are  like 
portions  of  a  unit  in  the  multiplier. 

Multiplying  by  -J-,  is  taking  1  half  of  the  multiplicand 
once.  Thus,  6xi=3.  (Art.  104.  Obs:) 

Multiplying  by  -J-,  is  taking  1  third  of  the  multiplicand 
once.  Thus,  6xi=2. 

Multiplying  by  -f,  is  taking  1  third  of  the  multiplicand 
twee.  Thus,  6x-f=4.  * 

OBS.  If  the  multiplier  is  a  unit,  the  product  is  equal  to  the  multi- 
plicand; if  the  multiplier  is  greater  than  a  unit,  the  product  is  greater 
than  the  multiplicand;  (Art.  45;)  and  if  the  multiplier  is  less  than  a 
unit,  the  product  is  less  than  the  multiplicand. 

EXERCISES   FOR   THE   SLATE. 
CASE   I. 

11.  If  a  bushel  of  corn  is  worth  £  of  a  dollar,  how  much 
as  5  bushels  worth  ? 


QUEST. — 131.  What  is  meant  by  multiplying  by  a  whole  number  ? 
132.  By  a  fraction?  Byi?  By  i?  By  -ft  Byf?  By!?  06s.  If 
the  multiplier  is  a  unit  or  1,  what  is  the  product  equal  to  ?  When  the 
multiplier  is  greater  than  1,  how  is  the  product,  compared  with  the 
multiplicand  ?  When  less,  how  ? 


122  MULTIPLICATION   OF  [SECT.  VL 

Solution.  —  5  bushels  will  cost  5  times  as  much  as  1 
bushel.  Now  ix5=-f,  or  2-J-  ;  that  is,  5  times  i  are  5 
halves,  equal  to  2  and  1  half.  Ans.  2-J-  dollars. 


12.  Multiply  -f-  by  5.  Ans^  or  3f. 

13.  Multiply  T^  by  8.  14.  Multiply  -£  by  12. 
15.  Multiply  A  by  18.           16.  Multiply  if  by  10. 

17.  If  a  pound  of  tea  cost  6  shillings,  how  much  will 
f  of  a  pound  cost  ? 

Solution.  —  Multiplying  by  -f  ,  is  taking  1  third  oi  the 
multiplicand  twice.  (Art.  132.)  Now  1  third  of  6  is  the 
same  as  6  thirds  of  1,  or  f  ;  and  2  thirds  of  6  must  be  2 
times  as  much  ;  that  is,  f  x2=V  5  and  J§a=4.  A/is. 


Note. — Since  the  product  of  any  two  numbers  will  be  the  same, 
whichever  is  taken  for  the  multiplier,  (Art.  47,)  the  fraction  may  bo 
taken  for  the  multiplicand,  and  the  whole  number  for  the  multiplier, 
when  it  is  more  convenient. 

Thus,  -f  X6=Y,  or  4  ;  and  6x1=4. 

18.  Multiply  12  by  -J-.  Ans.  3. 

19.  Multiply  .10  by  f.  20.  Multiply  15  by  f. 
21.  Multiply  -f  by  2.  Ans.  •fx2=J^L,  or  1|. 

Suggestion. — Dividing  the  denominator  of  a  fraction  b* 
any  number,  multiplies  the  value  of  the  fraction  by  thai 
number.  (Art.  114.)  Now,  if  we  divide  the  denominator 
8  by  2,  the  fraction  will  become  -f,  which  is  equal  to  1-J-, 
the  same  as  before.  Hence, 

133.  To  multiply  a  fraction  and  a  whole  number 
together. 

Multiply  the  numerator  of  the  fraction  by  the  whole  number, 
a-nd  write  the  product  over  the  denominator. 

Or,  divide  the  denominator  by  the  whole  number,  when  this 
vin  be  done  without  a  remainder.  (Art.  1 14.) 

QUEST. — 133.  How  multiply  a  fraction  and  a  whole  number  together! 


ARTS.   133-134.  a.]        FRACTIONS.  123 

OBS.  1.  A  fraction  is  multiplied  into  a  number  equal  to  its  denomi- 
nator by  canceling  the  denominator.  (Arts.  89,  91.)  Thus  -f-X7=  4. 

2.  On  the  same  principle,  a  fraction  is  multiplied  into  any  factor 
in  its  denominator,  by  canceling  that  factor.  (Arts.  91,  114.)  Thus, 


22.  Multiply  if-  by  5.  Ans.  *£-,  or  3. 

23.  Multiply  f£  by  9.  24.  Multiply  if  by  25. 
25.  Multiply  36  by  ff.  26.  Multiply  120  by  if. 
27.  Multiply  fff  by  25.  28.  Multiply  ff£  by  50. 

29.  Multiply  9i  by  5. 

Operation.        5  times  i  are  •£,  which  are  equal  to  2 
9i        and  £.     Set  down  the  £.    5  times  9  are  45, 
and  2  (which  arose  from  the  fraction)  make 
Ans.  47£        47.     Hence, 

134.  To  multiply  a  mixed  number  by  a  whole  one. 

Multiply  the  fractional  part  and  the  whole  number  sepa- 
ately,  and  unite  the  products. 

30.  Multiply  15f  by  7.  Ans.   110J-. 

31.  Multiply  25-f-  by  10.       32.  Multiply  48-Lg-  by  8. 

33.  Multiply  24  by  3£. 

Operation.  We  first  multiply  24  by  3,  then  by  -£, 

2)24  and  the  sum  of  the  products  is  84.     Mul- 

3^  tiplying  by  •£  is  taking  one  half  of  the  mul- 

72  tipiicand  once.     (Art.  132.)     But  to  find  a 

12  half  of  any  numbe"   we  divide  the  num- 

ber by  2.     (Art.  104.  Obs.)     Hence, 
/ins.  o  4 

134*  a.  To  multiply  a  whole  by  &  mixed  number. 

Multiply  first  by  the  integer,  then  by  the  fraction,  &nd  add 
\*e  products  together. 

34.  Multiply  27  by  3£.  Ans.  90. 

35.  Multiply  63  by  lOf        36.  Multiply  75  by  !2f 

QUEST.  —  Obs.  How  is  a  fraction  multiplied  by  a  number  equal  to 
Its  denominator  ?  How  by  any  factor  in  its  denominator  ?  134.  How 
i»  a  mixed  number  multiplied  by  a  whole  one  ?  134.  a.  How  i&  a 
whole  number  multiplied  by  a  mired  number  ? 


124  MULTIPLICATION   OF  [SECT.    VI, 


CASE    II. 

37.  A  man  owning  •§•  of  a  ship,  sold  -f  of  what  he 
owned.     What  part  of  the  ship  did  he  sell  ? 

Analysis.  —  J-  of  -f  is  -ft-  ;  for,  multiplying  the  denomi* 
nator  by  any  number,  divides  the  value  of  the  fraction. 
(Art.  1  13.)  Now  2  thirds  of  f  is  twice  as  much  ;  that  is, 
=-i65,  which,  reduced  to  its  lowest  terms,  is  •£.  Ans. 


Or,  we  may  reason  thus  :  Since  he  owned  f  ,  and  sold 
f  of  what  he  owned,  he  must  have  sold  -J  of  -f  of  the 
ship.  Now  -f  of  f  is  a  compound  fraction,  whose  value 
is  found  by  multiplying  the  numerators  together  for  a 
new  numerator,  and  the  denominators  for  a  new  denomi- 
nator.  (Art.  123.) 


Solution.  —  f  xi=Aj  or  f  •  An*-     Hence, 
135*  To  multiply  a  fraction  by  a  fraction. 

Multiply  the  numerators  together  for  a  new  numerator  and 
the  denominators  together  for  a  new  denominator. 

OBS.  It  will  be  seen  that  the  process  of  multiplying  one  fraction  by 
another,  is  precisely  the  same  as  that  of  reducing  compound  fractions 
to  simple  ones. 

38.  Multiply  i  by  f  .  Ans.  •&=£. 

39.  Multiply  f  by  f  .  40.  Multiply  f  by  f  . 
41.  Multiply  •&  by  f.           42.  Multiply  H  by  -f. 
43.  Multiply  -f  and  -f  and  -f  and  •£•  together. 

Operation. 

123^2  Since  the  factors  3  and  4  are  common 
-X-X-X-=  —  both  to  the  numerators  and  denomina- 
7  $  ^  5  35  tors,  we  may  cancel  them,  and  multiply 
the  remaining  factors  together  as  in  reducing  compound 
fractions  to  simple  ones.  (Art.  124.)  Hence, 


QUEST.— 135.  How  is  a  fraction  multiplied  by  a  fraction  ?  Obs.  To 
what  is  the  process  of  multiplying  one  fraction  by  another  cimilar  I 
136-  How  multiply  fractions  together  by  cancelation  ? 


ARTS.  135-137.]  FRACTIONS.  125 

136*  To  multiply  fractions  by  CANCELATION. 

Cancel  all  the  factors  common  both  to  the  numerators  and 
denominators  ;  then  multiply  the  remaining  factors  in  the 
numerators  together  for  a  new  numerator  ',  and  those  remain- 
ing in  the  denominators  for  a  new  denominator^  as  in  reduc- 
tion of  compound  fractions.  (Art.  124.) 

OBS.  1.  This  process,  in  effect,  is  dividing  the  product  of  the  nu- 
merators and  that  of  the  denominators  by  the  same  number,  and 
therefore  does  not  alter  the  value  of  the  answer.  (Art.  116.) 

2.  Care  must  be  taken  that  the  factors  canceled  in  the  numerators 
are  exactly  equal  to  those  canceled  in  the  denominators. 

44.  Multij   /  -f-  by  £.     Ans.  f  . 

45.  Multij   y  -f  by  i  and  f.     Ans.  rV 

46.  Multi]   y  -fe  by  -f-  and  f. 

47.  Multiply  f  by  -^  and  -J-  and  -}-f  . 

48.  Multiply  -f^  by  if  and  £  and  f  and  f  . 

49.  Multiply  f-f-  by  -ft-  and  -fa  and  -^  and  f  . 

50.  Multiply  7£  by  3^. 

Solution.  —  7-J-,  reduced  to  an  improper  fraction,  be- 
comes J^,  and  3i  becomes  -^  Now  -^X^-^-1^,  or 

25. 


137*  Hence,  when  the  multiplier  and  multiplicand 
are  both  mixed  numbers,  they  should  be  reduced  to  im- 
proper fractions,  and  then  be  multiplied  according  to  the 
rule  above. 


EXAMPLES   FOR   PRACTICE. 


1.  What  will  12  apples  cost,  at  •£  of  a  cent  apiece  ? 

2.  If  a  bushel  of  wheat  weighs  f  of  a  hundred  weight, 
how  much  will  10  bushels  weigh  ? 

3.  If  a  man  earns  -f-  of  a  dollar  per  day,  how  much 
can  he  earn  in  12  days  ? 


QUEST. — Ohs.  How  does  it  appear  that  this  process  will  give  the 
true  answer  ?  What  is  necessary  to  be  observed  with  regard  to  can- 
celing factors  ?  137.  When  the  multiplier  and  multiplicand  are  mixed 
numbers,  how  proceed  ? 


126  MULTIPLICATION   OP  [SECT,   VL 

4.  If  a  family  consume  f  of  a  barrel  of  flour  in  a  week, 
how  much  wiil  they  consume  in  15  weeks? 

5.  If  I  burn  •£  of  a  cord  of  wood  in  a  month,  how  much 
shall  I  bum  in  12  months? 

6.  If  a  man  can  reap  -^  of  an  acre  of  grain  in  a  day 
how  many  acres  can  he  reap  in  9  days? 

7.  If  a  pound  of  powder  is  worth  6  shillings  how  much 
is  -f  of  a  pound  worth  ? 

8.  If  a  gallon  of  oil  is  worth  7  shillings,  how  much  is 
5-  of  a  gallon  worth  ? 

9.  When  beeric1  10  dollars  a  barrel,  how  much  will  i 
of  a  barrel  cost  ? 

10.  What  will  i  of  a  firkin  of  butter  cost,       15  dollars 
a  firkin  ? 

11.  At  f  of  a  dollar  a  cord,  how  much  wii'.  the  sawing 
of  20  cords  of  wood  amount  to  ? 

12.  What  will   16  pounds  of  cheese  cost,  at  8£  cents 
per  pound  ? 

13.  Wh- 1  cost  9  dozen  of  eggs,  at  12£  cents  per  dozen? 

14.  What  cost  15-f  yards  of  cambric,  at  15  pence  per 
yard? 

15.  What  cost  1  !•£  cords  of  wood,  at  1£  dollar  per  cord  ? 

16.  At  12  cents  a  pound,  what  cost  2-f  pounds  of  pep- 
per? 

17.  At  5  shillings  a  pound,  what  cost  12f  pounds  of  tea  ? 

18.  What  will  6  pounds  of  starch  come  to,  at  J2-£  cents 
per  pound  ? 

19.  What  will  18  ounces  of  nutmegs  come  to,  at  6i 
cents  an  ounce? 

20.  At  12-f  cents  a  yard,  what  will  17  yards  of  cotton 
come  to  ? 

21.  At  3^  dollars  a  yard,  what  cost  15  yards  of  broad- 
cloth ? 

22.  What  cost   15f-  yards  of  ribbon,  at  10  cents  per 
yard? 

23.  What  cost  22  pocket  handkerchiefs,  at  if  of  a 
dollar  apiece? 

24.  At  -fo  of  a  dollar  a  yard, what  will  -f-  of  a  yard  of  lac« 
cost? 


ART.  137.]  FRACTIONS.  127 

25.  At  f  of  a  dollar  a  yard,  what  will  -f  of  a  yard  of 
muslin  come  to  ? 

26.  At  f  of  a  dollar  a  bushel,  what  cost  T97  of  a  bushel 
of  wheat  ? 

27.  What  will  f  of  a  pound  of  tea  cost,  at  -f  of  a  dollar 
a  pound  f 

28.  What  cost  66  bushels  of  apples,  at  18f  cents  a 
bushel  ? 

29.  At  62£  cents  a  yard,  what  cost  12£  yards  of  balzo- 
rine? 

30.  What  cost  18£  yards  of  tape,  at  6-J-  cents  per  yard  ? 

31.  What  cost  13  bushels  of  oats,  at  18f  cents  per 
bushel? 

32.  What  cost  31£  yards  of  sheeting,  at  ^  of  a  dollar 
per  yard  ? 

33.  At  T^  of  a  dollar  a  quart,  what  cost  8$  quarts  of 
cherries  ? 

34.  At  3-f-  shillings  a  yard,  what  cost  7$  yards  of  ging- 
ham? 

35.  What  cost  14f  bushels  of  potatoes,  at  18^-  cents  a 
bushel  ? 

36.  At  7-f  shillings  a  yard,  what  cost  8-f  yards  of  silk  1 

37.  At  -I  of  a  dollar  a  bushel,  what  cost  47-f-  bushels  of 
peaches  ? 

38.  What  cost  63^-  pounds  of  sugar,  at  9-f-  cents  per 
pound  ? 

39.  What  cost  2f  yards  of  velvet, at  3-f  dolls. a  yard? 

40.  What  cost  9f  yards  of  calico,  at  1-f  shillings  a  yard  ? 

41.  WKat  cost  25-|-  pounds  of  figs,  at  15£  cents  a  pound  ? 

42.  What  cost  35f  cords  of  wood,  at  18-J-  shillings  per 
cord? 

43.  What  cost  175£  bushels  of  corn,  at  -f  of  a  dollar  a 
bushel? 

44.  What  cost  8f  tons  of  hay,  at  15£  dollars  a  ton  ? 

45.  If  a  man  can  travel  42£  miles  in  one  day,  how  fnr 
ean  he  travel  in  17£  days? 


128  DIVISION  OP  [SECT.  VL 


DIVISION  OF  FRACTIONS. 

MENTAL     EXERCISES. 

Ex.  1.  A  man  divided  f  of  a  pound  of  honey  equally 
among  his  3  children :  what  part  of  a  pound  did  each 
receive  ? 

Analysis. — 1  is  one  third  of  3  ;  therefore  1  child  must 
have  received  1  third  of  6  sevenths.  1  third  of  6  sevenths 
is  2  sevenths.  Ans.  Each  child  received  f  of  a  pound. 

2.  If  4  pounds  of  loaf  sugar  cost  -f  of  a  dollar,  how 
much  will    1    pound  cost  ? 

3.  A  father  gave  his  2  sons  if  of  a  dollar :  how  many 
twelfths  did  each  receive  ? 

4.  A  little  girl  bought  5  lead  pencils  for  if  of  a  shil 
ling :  how  much  did  she  give  apiece  for  them  ? 

5.  A  father  gave  •§•£  parts  of  a  vessel  to  his  6  sons  • 
what  part  of  the  vessel  did  each  receive? 

6.  At  -J-  dollar  a  yard,  how  many  yards  of  French  mus- 
lin can  you  buy  for  4  dollars  ? 

Suggestion. — 4  dollars  will  buy  as  many  yards  as  1 
half  is  contained  times  in  4,  or  as  there  are  halves  in  4 
dollars.  Now  since  there  are  2  halves  in  1  dollar,  in  4 
dollars  there  are  4  times  2  halves  ;  and  4  times  2  halves 
are  8  halves.  Ans.  4  dollars  will  buy  8  yards. 

7.  At  •£  cent  apiece,  how  many  apples  can  I  buy  for 
6  cents? 

8.  At  •£  of  a  dollar  a  pound,  how  many  pounds  of 
aimonds  can  you  buy  for  12  dollars? 

9.  How  many  quills,  at  f  of  a  penny  apiece,   can  you 
buy  for  f  of  a,  penny  ? 

Suggestion. — f  of  a  penny  will  buy  as  many  quills  as 
£  is  contained  times  in  f ;  and  -f  is  contained  in  f ,  3  times. 

Ans.  3  quills. 

10.  How  many  yards  of  cloth  can  I  buy  for  £  of  a  cord 
of  wood,  if  I  give  \  of  a  cord  for  a  yard  of  cloth  ? 


ART.  138.]  FRACTIONS.  129 


EXERCISES   FOR    THE   SLATE. 
CASE  I. 

11.  If  3  bushels  of  oats  cost  f  of  a  dollar,  what  will  1 
bushel  cost? 

Analysis. — 1  is  1  third  of  3 ;  therefore,  1  bushel  will 
cost  1  third  part  as  much  as  3  bushels.  1  third  of  f  is  £ . 

Ans.  i  of  a  dollar. 

We  divide  the  numerator  of  the  frac- 
Operation.       tion  -f ,  whicl^  is  the  whole  cost,  by  3  the 
|-s-3=|.  Ans.     whole  number  of  bushels,  and  place  the 
quotient  2  over  the  given  denominator. 

12.  If  4  yards  of  calico  cost  -f-  of  a  dollar,  what  will  1 
yard  cost  ? 

Operation.  In  this  case  we  cannot  divide 

f  -*-4=*&r,  or  A-  Ans.     the  numerator  of  the  dividend  by 

4  the  given  divisor,  without  a 

remainder.  We  therefore  multiply  the  denominator  by 
the  4,  which  is  in  effect  dividing  the  fraction.  (Art.  113.) 
Hence, 

138*  To  divide  a  fraction  by  a  whole  number 

Divide  the  numerator  by  the  whole  number,  when  it  can  be 
done  without  a  remainder;  but  when  this  cannot  be  done, 
multiply  the  denominator  by  the  whole  number. 

13.  Divide  f  by  3. 

First  Method.  Second  Method. 

£-5-3=1,  or  \.  Ans.  f  -*-3=^r,  or  \.  Ans. 

14.  Divide  H  by  6.  15.  Divide  if  by  8. 
16.  Divide  -Vs-  by  7.  17.  Divide  H  by  12. 


QVEST.-—138.  How  is  a  fraclioa  divided  by  a  whole  number  t 
5 


130  DIVISION  OF  [SECT.  VL 

18.  Pivide  fg-  by  9.  19.  Divide  If  by  8. 

20,  Divide  Hi  by  25.  21.  Divide  -Hi  by  30. 


CASE  II. 

22.  At  -J-  of  a  dollar  a  pound,  how  many  pounds  of 
honey  can  be  bought  for  f  of  a  dollar  ? 

Suggestion.  —  Since  •}•  of  a  dollar  will  buy  1  pound,  f  oi 
a  dollar  will  buy  as  many  pounds  as  \  is  contained  times 
in  -f.  Now  \  is  contained  in  f  ,  3  times.  Ans.  3  pounds. 

23.  At  £  of  a  dollar  a  bushel,  how  much  barley  can  be 
bought  for  f  of  a  dollar  ? 

We   first    reduce    the    given 

First  Operation.  fractions  to  a  common  denomi- 

f=-jHJ-  nator;  (Art.    125;)  then  divide 

i=-fo  the  numerator  of  the  dividend 

-^B-f.^=l^..    Ans,     by  the  numerator  of  the  divisor, 

as  above. 

OBS.  1.  After  the  fractions  are  reduced  to  a  common  denominator, 
it  will  be  perceived  that  no  use  is  made  of  the  common  denominator 
itself.  In  practice,  therefore,  it  is  simply  necessary  to  multiply  the 
numerator  of  the  dividend  by  the  denominator  of  the  divisor,  and  the 
denominator  of  the  dividend  by  the  numerator  of  the  divisor,  in  the 
same  manner  as  two  fractions  are  reduced  to  a  common  denominator; 
or,  what  is  the  same  in  effect,  invert  the  divisor,  and  proceed  as  in 
multiplication  of  fractions.  (Art.  135.) 

Note.  —  To  invert  a  fraction  is  to  put  the  numerator  in  the  place  of 
the  denominator,  and  the  denominator  in  the  place  of  the  numerator, 
Thus,  in  the  example  above,  inverting  the  divisor  -f  ,  it  becomes  f  j 
and  fXf  —  *£-,  or  1^,  which  is  the  same  as  before. 

Again,  we  may  also  illustrate  the  principle  thus  : 

Second  Operation.  Dividing  the  dividend  f  by  2,  the 
•f~t-2=-f  quotient  is  f.  (Art.  113.)  But  it  is 

|X5=  -^  required  to  divide-  it  by  only  •£  of  2  ; 

And-^-=l-£.  Ans.  consequently  the  -f  is  5  times  too 
small  for  the  true  quotient.  There- 

fore -f  multiplied  by  5  will  be  the  quotient  required. 

Now  $xo—  ^,  or  1-;,  which  is  the  same  result  as  before, 


ART.  i39.J  FRACTIONS.  131 

OBS.  2.  By  examination  the  learner  will  perceive  that  this  process 
is  precisely  the  same  in  effect  as  the  preceding ;  for  in  both  cases  the 
denominator  of  the  dividend  is  multiplied  by  the  numerator  of  the  di- 
visor, and  the  numerator  of  the  dividend,  by  the  denominator  of  the 
divisor.  Hence, 

139.  To  divide  a  fraction  by  a  fraction. 

I.  If  the  given  fractions  have  a  common  denominator; 

Divide  the  numerator  of  the  dividend  by  the  numerator  of 
the  divisor. 

II.  When  the  fractions  have  not  a  common  denominator  ; 

Invert  the  divisor,  and  proceed  as  in  multiplication  of  frac- 
tions. (Art  135.) 

OBS.  1 .  Compound  fractions  occurring  in  the  divisor  or  dividend, 
must  be  reduced  to  simple  ones,  and  mixed  numbers  to  improper 
fractions. 

2.  The  method  of  dividing  a  fraction  by  a  fraction  depends  upon 
the  obvious  principle,  that  if  two  fractions  have  a  common  denomi- 
nator, the  numerator  of  the  dividend,  divided  by  the  numerator  of  the 
divisor,  will  give  the  true  quotient.  Now  multiplying  the  numerator 
of  the  dividend  by  the  denominator  of  the  divisor,  and  the  denomi- 
nator of  the  dividend  by  the  numerator  of  the  divisor,  is  in  effect  re- 
ducing the  two  fractions  to  a  common  denominator.  The  object  of 
inverting  the  divisor,  is  simply  for  convenience  in  multiplying. 

24.  Divide  |  off  by  1£. 

Solution. — |  of  $=*&,  and  l|=f     Now 
or  £f .  Ans. 

25.  Divide  7-J-  by  2|.  Ans.  3*. 

26.  Divide  13*-  by  f.  27.  Divide  -f-  by  1-f. 
28.  Divide  ff-  by  -J-f  29.  Divide  -f-f  by  £ 


QUEST. — 139.  How  is  one  fraction  divided  by  another  when  they 
have  a  common  denominator  ?  How,  when  they  have  not  common 
denominators  ?  Ohs.  How  proceed  when  the  divisor  or  dividend  are 
compound  fractions,  or  mixed  numbers  ?  Upon  what  principle  doea 
the  method  of  dividing  a  fraction  by  a  fraction,  depend  ?  Why  mul- 
tiply the  numerator  of  the  dividend  by  the  denominator  of  the  divisor 
&c.  ?  Why  invert  the  divisor  ? 


182  DIVISION  OF  -[SECT.  VI, 

30.  Divide  i  of  $  by  •£  off. 

Operation.  For  convenience  we  arrange  th* 

numerators,  (which  answer  to  divi- 


* dends,)  on  the  right  of  a  perpendic- 

ular line,  and  the  denominators, 
(which  answer  to  divisors,)  on  the 
le 


left ;  then  canceling  the  factors  3 
8  I  5=-|.  Ans.  and  2,  which  are  common  to  both 

sides,  (Art.  91.  a,)  we  multiply  the 
remaining  factors  in  the  numerators  together,  and  those 
remaining  in  the  denominators,  as  in  the  rule  above. 
Hence, 

1 4O*  To  divide  fractions  by  CANCELATION. 

Having  inverted  the  divisor,  cancel  all  the  factors  common 
both  to  the  numerators  and  denominators,  a?id  proceed  as  in 
multiplication  of  fractions.  (Art.  136.) 

OBS.  Before  arranging  the  terms  of  the  divisor  for  cancelation,  it 
is  always  necessary  to  invert  them,  or  auppose  them  to  be  inverted. 

31.  Divide  4£  by  2i.  Ans.  2. 

32.  Divide  f  of  6  by  |  of  4.  33.  Divide  4|  by  i  of  ^. 
34.  Divide-f  offfby^of  f  35.  Divide  -&  of  f  by  f 
36.  Divide  i  of  15-f-  by  4-f.  37.  Divide  i  by  ff  of  -fr. 
38.  Divide  f  by  -ft  of  2f  39.  Divide  25i  by  i  of  26, 

CASE  III. 

40.  A  merchant  sent  12  barrels  of  flour  to  supply  some 
destitute  people,  allowing  -f  of  a  barrel  to  each  family. 
How  many  families  shared  in  his  bounty  ? 

Solution. — If -f  of  a  barrel  supplied  1  family,  12  barrels 
will  supply  as  many  families  as  -f  is  contained  times  in  12. 
Reducing  the  dividend  12  to  the  form  of  a  fraction,  it  be- 
comes  -V*;  now  inverting  the  divisor,  we  have  -^X^—3/- 
or  18.  Ans.  18  families. 

QUEST.-— 140.  How  divide  fractions  by  cancelation  ?  How  arranga 
the  terms  of  the  given  fractions  ?  Oba.  What  must  be  done  to  the  divi- 
»or  before  arranging  its  terms  ? 


ART.  140-143.]  FRACTIONS.  133 

Or,  we  may  reason  thus  :  -£•  is  contained  in  12,  as  many 
times  as  there  are  thirds  in  1  2,  viz  :  36  times.  Now  2 
thirds  are  contained  in  12,  only  half  as  many  times  as  1 
third;  and  36-=-2=18.  Ans.  Hence, 

141*  To  divide  a  whole  number  by  a  fraction. 

Reduce  the  whole  number  to  the  form  of  a  ft  -action  ,  (Art. 
122.  Obs.  1,)  and  then  proceed  according  to  the  rule  for  di- 
viding a  fraction  by  a  fraction.  (Art.  139.) 

Or,  multiply  the  whole  number  by  the  denominator,  ana 
divide  the  product  by  the  numerator. 

OBS.  When  the  divisor  is  a  mixed  number,  it  must  be  reduced  to 
an  improper  fraction,  then  proceed  as  above. 

41.  Divide  120  by  3f.  Ans.  33i. 

42.  Divide  35  by  •£.  43.  Divide  47  by  f 
44.  Divide  165  by  f             45.  Divide  237  by  4  \. 

142.  From  the  definition  of  complex  fractions,  and 
the  manner  of  expressing  them,  it  will  be  seen  that  they 
arise  from  division  of  fractions.  Thus  the  complex  frac- 

41 
tion  —  |,  is  the  same  as  -f-s-f-  ;  for,  the  numerator  4£=-f, 

and  the  denominator  l-J-=f;  but  the  numerator  of  a  frac- 
tion is  a  dividend,  and  the  denominator  a  divisor.  (Art. 
109.)  Now  -f-HHf  $,  which  is  a  simple  fraction.  Hence, 

143*  To  reduce  a  complex  fraction  to  a  simple  one. 

Consider  the  denominator  as  a  divisor,  and  proceed  as  in 
division  of  fractions.  (Art.  139.) 

2_L 

46.  Reduce  —  to  a  simple  fraction. 
5f 

Operation. 

Now  f^/H^fa  or  ff  .  Ans. 


k    \ 


QUEST. — 141.'  How  is  a  whole  number  divided  by  a  fraction?  Oba, 
How  by  a  mixed  number?  142.  From  what  do  complex  fraction* 
irise  ?  143.  How  reduce  them  to  simple  fractions  ? 


134  DIVISION  OP  [SECT.  VI 

47.  Reduce  to  a  simple  fraction.  Ans.  JA 

3i 

48.  Reduce  _I  to  a  simple  fraction.  A?is.  ft. 

6 

f 

49.  Reduce  ^  to  a  simple  fraction.  Ans.  -ft. 

50.  Reduce  the  following  complex  fractions  to  simple 
ones. 

4-|      8      9-J-     12-J-     18-J-     20-i- 

T    5f    7i    "6f    T2i     25l 
144*  To  multiply  complex  fractions  together. 

First  reduce  the  complex  fractions  to  simple  ones  ;  (Art, 
143  ;)  then  arrange  the  terms,  and  cancel  the  commo?i  factors 
as  in  multiplication  of  simple  fractions.  (Art.  136.) 

OBS.  1 .  The  terms  of  the  complex  .  /actions  may  be  arranged  for 
reducing  them  to  simple  ones,  and  for  multiplication  at  the  same  time. 

2.  To  divide  one  complex  fraction  by  another,  reduce  them  to  sim 
pie  fractions,  then  proceed  as  in  Art.  139. 

51.  Multiply^  by  || 

Operation.  The    numerator    2-£— f.     (Art. 

122.)     Place  the  7  on  the  right 

Q 

0 

* 

(Art.  143;)  i.  e.  place  the  4  on  the 
3  |  8=2|.  Ans.  right  and  the  9  on  the  left  of  the 

line.  4£=£,  and  lf=f,  both  oi 

which  must  be  arranged  in  the  same  manner  as  the  terms 
of  the  multiplicand.  Now,  canceling  the  common  fac- 
tors, we  divide  the  product  of  those  remaining  on  the 
right  of  the  line  by  the  product  of  those  on  the  left,  and 
the  quotient  is  2|.  (Art.  136.) 

QUEST. — 144.  How  are  complex  fractions  multiplied  together  I  Ob* 
How  is  one  complex  fraction  divided  by  another  ? 


hand  and  3  on  the  left  of  the  per- 
4  pendicular  line.     The  denomina- 

tor 2f=f  5  which  must  be  inverted ; 


ART.  144.]  FRACTIONS.  135 

52    Multiply   ?iby?t         53.  Multiply  fUby??. 
2-f       A  6i    *21 

54.  Multiply      xbyl.      65.  Multiply       xb 


EXAMPLES   FOR   PRACTICE. 

1.  At  %  dollar  per  bushel,  how  many  bushels  of  pears 
can  be  bought  for  5  dollars  ? 

2.  At  -f-  of  a  penny  apiece,  how  many  apples  can  be 
bought  for  18  pence? 

3.  At  f  of  a  dollar  a  pound,  how  many  pounds  of  tea 
will  7  dollars  buy  ? 

4.  How  many  bushels  of  pears,  at  1£  dollar  a  bushel, 
can  be  purchased  for  15  dollars? 

5.  How  many  gallons  of  molasses,  at  2£  dimes  per 
gallon,  will  10  dimes  buy? 

6.  How  many  yards  of  satinet,  at   If  of  a  dollar  per 
yard,  can  be  purchased  for  20  dollars  ? 

7.  At  4-f  dollars  per  yard,  how  many  yards  of  cloth 
can  be  obtained  for  25£  dollars  ? 

8.  At  6f  cents  a  mile,  how  far  can  you  ride  for  62£ 
cents  ? 

9.  At  12£  cents  a  pound,  how  many  pounds  of  flax 
will  67-f-  cents  buy  ? 

10.  At  16-J-  cents  per  pound,  how  many  pounds  of  figs 
can  you  buy  for  87^-  cents  ? 

11.  How  many  cords  of  wood,  at  6£  dollars  per  cord, 
will  it  take  to  pay  a  debt  of  Q7$  dollars  ? 

12.  How  many  barrels  of  beer,  at  llf  dollars  per  bar- 
rel, can  be  obtained  for  95-J-  dollars? 

13.  A  man  bought  15-|  barrels  of  beef  for  124f  dollars, 
how  much  did  he  give  per  barrel  ? 

14.  A  man  bought  13£  pounds  of  sugar  for  94£  cents: 
Aow  much  did  his  sugar  cost  him  a  pound  ? 

15.  A  lady  bought   15-f  yards  of  silk  for   145A  shil- 
lings :  how  much  did  she  pay  per  yard  ? 

16.  Bought   151  baskets  of  peaches  for  24|  dollars  : 
how  much  was  the  cost  per  basket  ? 


136      ,  COMPOUND  [SECT.  VII 


17.  Bought  30i  yards  of  broadcloth,  for  181-£  dollars, 
what  was  the  price  per  yard  ? 

18.  Paid  375  dollars  for  125|  pounds  of  indigo  :  what 
was  the  cost  per  pound  ? 

19.  How  many  tons  of  hay,  at   16-£  dollars  per  ton, 
can  be  bought  for  196^-  dollars  ? 

20.  How  many  sacks  of  wool,  at  17  i  dollars  per  sack, 
can  be  purchased  for  1500  dollars  ? 

21.  How  many  bales  of  cotton,  at  15-f  dollars  per  bale, 
can  be  bought  for  2500  dollars  ? 

22.  Divide  145^  by  16.          23.  Divide  16ft  by  25. 
24.  Divide  8526  by  45^.        25.  Divide  12563  by  68^- 
26.  Divide  85ff  by  18$.          27.  Divide  105^-  by  82-&. 
28.  Divide  f  of  -fa  by  6£.       29.  Divide  -f  of  16  by  -f  off. 
30.  Divide  -ft  of  30  by  19.      31.  Divide  f  of  -f  by  21. 
32.  Divide  T\of  f£  by  f  of  31.  33.  Divide  -^of 


SECTION    VII. 
COMPOUND    NUMBERS. 

AUT.  146.  Numbers  which  express  things  of  the 
same  kind  or  denomination,  as  3  pears,  7  rose?,  15  horses, 
are  called  simple  numbers. 

Numbers  which  express  things  of  different  kinds  or  de- 
nominations, as  the  divisions  of  money,  weight,  and  mea- 
sure, are  called  compound  numbers.  Thus  6  shillings  7 
pence ;  5  pounds  2  ounces ;  7  feet  3  inches,  &c.,  are 
compound  nnmbers. 

OBS.  Compound  Numbers,  by  some  late  authors,  are  called  De- 
nominate Numbers. 


QUEST. — 146.  What  are  simple  numbers  ?     What  are  compound 
numbers  * 


ARTS,  146,  147.]  NUMBERS,  137 

/ 

STERLING  MONEY. 

147.  Sterling  Money  is  the  national  currency  of 
England. 

4  farthings  (qr,  or  far.)  make  1  penny,  marked         d. 
12  pence  "      1  shilling,     "  s. 

20  shillings  "      1  pound,  or  sovereign,£. 

21  shillings  "      1  guinea. 

OBS.  1.  It  is  customary,  at  the  present  day,  to  express  farthings  in 
fractions  of  a  penny.  Thus,  1  qr.  is  written  i  d.;  2  qrs.,-1-  d.;  3  qrs. 
ad. 

2.  The  Pound  Sterling  is  represented  by  a  gold  coin,  called  a 
Sovereign.  According  to  Act  of  Congress,  1842,  it  is  equal  to  4 
Dollars  and  8-1  cents. 

MENTAL    EXERCISES. 

1.  In  5  pence,  how  many  farthings? 

Solution. — Since  there  are  4  farthings  in  1  penny,  in  5 
pence  there  are  5  times  as  many ;  and  5  times  4  are  20. 

Ans.  20  farthings. 

2.  In  8  pence,  how  many  farthings  ?  In  10d.?  In  12d.  ? 

3.  How  many  shillings  are  there  in  3  pounds?     In 
£5?     In  £8?     In  £10? 

4.  How  many  pence  are  there  in  8  farthings? 

Solution. — Since  4  farthings  make  1  penny,  8  farthings 
will  make  as  many  pence,  as  4  is  contained  times  in  8 ; 
and  4  is  contained  in  8,  2  times.  Ans.  2  pence. 

5.  How  many  pence  in  12  farthings?  In  15  qrs.  ?    In 
20  qrs.  ?     In  25  qrs.  ?     In  33  qrs.  ?     In  36  qrs.  ? 

6.  How  many  shillings  in   15  pence?     In  24d.  ?     In 
30d.  ?     In  36d.  ?     In  60d.  ?     In  68d.  ?     In  75d.  ? 

7.  How  many  pounds  in  25  shillings  ?     In  30s.  ?     In 
40s.  ?     In  65s.  ?     In  80s.  ?     In  89s.  ? 


QUEST.— 147.  What  is  Sterling  Money  ?  Repeat  the  Table.  Obs. 
How  are  farthings  usually  expressed  ?  How  is  a  pound  sterling  repre- 
lented  ?  What  is  its  value  in  dollars  and  centa  ? 


198  COMPOUND  [SECT.  VIL 

TROY  WEIGHT. 

Note. — Most  children  have  very  erroneous  or  indistinct  ideas  of  th« 
weights  and  measures  in  common  use.  It  is,  therefore,  strongly  re- 
commended for  teachers  to  illustrate  them  practically,  by  referring  to 
some  visible  object  of  equal  magnitude,  or  by  exhibiting  the  ounce; 
the  pound;  the  linear  inch,  foot,  yard,  and  rod;  also  a  square  and 
cubic  inch,  foot,and  yard;  the  pint,  quart,  gallon,  peck,  bushel,  &c. 

/ 

148.  Troy  Weight  is  used  in  weighing  gold,  silver 
jewels,  liquors,  &c.,  and  is  generally  adopted  in  philo 
sophical  experiments. 

24  grains  (gr.)  make   1  pennyweight,  marked  pwt. 
20  pennyweights   "      1  ounce,  "         oz. 

12  ounces  "      1  pound,  "          Ib. 

OBS.  1.  The  standard  of  Weights  end  Measures  is  different  in  dif- 
ferent States  of  the  Union.  In  1834,  the  Government  of  the  United 
States  adopted  a  uniform  standard,  for  the  use  of  the  several  custoir 
houses  arid  other  purposes. 

2.  The  standard  unit  of  Weight  adopted  by  the  Government,  is  the 
Tray  Pound  of  the  United  States  Mint,  which  is  identical  with  the 
Imperial  Troy  pound  of  England,  established  by  Act  of  Parliament. 
A.  D.  1826.* 

3.  Troy  Weight  was  formerly  used  in  weighing  articles  of  every 


suppose 
think  it  was  derived  from  Troy-novant,  the  former  name  of  London.t 

8.  How  many  grains  in  2  pennyweights?  In  3  pwts? 
In  4  pwts  ? 

9.  How  many  pennyweights  in  2  ounces  ?     In  3  oz.  ? 
In  4  oz.  ?     In  5  oz.  ? 

1U.  How  many  ounces  in  2  pounds  ?     In  3  Ibs.  ?     In 
4  Ibs.  ?     In  5  Ibs.  ?     In  6  Ibs.  ?    In  7  Ibs.  ?     In  10  Ibs.  ? 

QUEST.— 148.  In  what  is  Troy  Weight  used  ?  Repeat  the  Table, 
Obs.  When  was  Troy  Weight  introduced  into  Europe  ?  From  what 
was  its  name  derived  ?  Do  all  the  States  have  the  same  standard  of 
weights  and  measures  ?  What  is  the  standard  unit  of  weight  adopted 
by  the  Government  of  the  United  States  ? 

*  Hassler  on  Weights  and  Measures,  p.  10.     Also,  Reports  of  the  Secretary  ol 
the  Treasury,  March  3,  1831 ;  and  June  20,  1832. 
t  Hind's  Arithmetic,  Art.  224.    Also,  North  America?  Review,  VoL  XLV 


ARTS.  148,  149.]  NUMBERS.  139 

AVOIRDUPOIS  WEIGHT. 

149*  Avoirdupois  Weight  is  used  in  weighing  gro- 
ceries and  all  coarse  articles  ;  as,  sugar,  tea,  coffee,  butter, 
cheese,  flour,  hay,  &c.,  and  all  metals,  except  gold  and 
silver. 

16  drams  (dr.)  make  1  ounce,  marked         oz. 

16  ounces  "      1  pound,       "  Ib. 

25  pounds  "      1  quarter,     "  qr. 

4  quarters, or  100  Ibs.  "      1  hundredweight,     cwt. 

20  hundred  weight         "      1  ton,  marked  T. 

OBS.  1.  The  Avoirdupois  Pound  of  the  United  Stales  is  determfced 
from  the  standard  Troy  Pound,  and  is  in  the  ratio  of  5760  to  7000  ;* 
that  is, 

1  pound  Troy          contains  5760  grains. 

1  pound  Avoirdupois     "        7000      "      Troy. 

1  ounce          "  "        437i     " 

Idram  "  "       27-T 


2.  The  British  Imperial  Pound  Avoirdupms  is  defined  to  be  the 
weight  of  27i7u2u7o4o  cubic  inches  of  distilled  water,  at  the  tempera- 
ture of  61°  Fahrenheit,  when  the  barometer  stands  at  30° .t 

3.  Gross  weight  is  the  weight  of  goods  with  the  boxes,  casks,  or 
bags  which  contain  them. 

Net  weight  is  the  weight  of  the  goods  only. 

4.  Formerly  it  was  the  custom  to  allow  112  pounds  for  a  hundred 
weight,  and  28  pounds  for  a  quarter ;  but  this  practice  has  become 
nearly  or  quite  obsolete.     In  buying  and  selling  all  articles  of  com- 
merce estimated  by  weight,  the  laws  of  most  of  the  States  as  w«U  as 
general  usage,  call  100  pounds  a  hundred  weight,  and  25  pounds  a 
quarter. 

11.  How  many  drams  are  there  in  2  ounces?     In  3 
oz.  ?     In  4  oz.  ?     In  5  oz.  ? 

12.  How  many  ounces  in  2  pounds?     In  3  Ibs.?     In 
4  Ibs.  ?     In  5  Ibs.  ? 

13.  How  many  pounds  in  2  quarters? 


QUEST. — 149.  In  what  is  Avoirdupois  Weight  used  \  Repeat  the 
Table.  Point  to  an  object  that  weighs  an  ounce.  A  pound.  Obs. 
How  is  the  Avoirdupois  pound  of  the  United  States  determined  1 
What  is  gross  weight  ?  Net  weight  I  How  many  pounds  were  for- 
merly allowed  for  a  hundred  weight  ?  For  a  quarter  1 

*  Reports  of  Secretary  of  Treasury,  March  3,  1832 :  June,  90,  1832.    Also, 
Congressional  Documents  of  1833. 
t  Hind's  Arithmetic,  Art.  223 


140  COMPOUND  [SECT.  VII, 

14.  How  many  quarters  in  2  hundred  weight?     In  3 
cwt.  ?     In  5  cwt.  ?     In  6  cwt.  ? 


APOTHECARIES'  WEIGHT. 

15O.  Apothecaries1    Weight  is  used  by  apothecaries 
and  physicians  in  mixing  medicines. 

20  grains  (gr.)  make  1  scruple,  marked  sc.,  or  3. 

3  scruples             "  1  dram,           "  dr.,  or  3. 

8  drams                "  1  ounce,          "  oz.:  or  £ . 

12  ounces               "  1  pound,         "  ft>. 

OBS.  I.  The  pound  and  ounce  in  this  weight  are  the  same,  as  thi 
Troy  pound  and  ounce ;  the  other  denominations  are  different. 
2.  Drugs  and  medicines  are  bought  and  sold  by  avoirdupois  weight 

15.  In  2  scruples,  how  many  grains?     In  3  sc. ? 

16.  In  3  drams,  how  many  scruples?     In  4  dr.?     In 
5  dr.?     In  7  dr.?* 

17.  In  2  pounds,  how  many  ounces  ?     In  3  as.  ? 

LONG  MEASURE. 

151*  Long  Measure  is  used  in  measuring  distances 
or  length  only,  without  regard  to  breadth  or  depth. 

12  inches  (in.}  make  1  foot,  marked     ft. 

1  yard,  "          yd. 

1  rod,  perch,  or  pole,       "  r.  or  p. 
1  furlong,  "       fur. 

1  mile,  m. 

I  league,  "  / 


3  feet 
51  yards,  or  16£  feet 
40  rods 

8  furlongs,  or  320  rods 
3  miles 


1 


360  degrees  make  a  great  circle,  or  the  circumference  of  the  earth. 

Note. — 4  inches  make  1  hand ;  9  inches,  1  span ;  18  inches,  J  cu- 
bit; 6  feet,  1  fathom. 


QUEST. — 150.  In  what  is  Apothecaries'  Weight  used  ?  Recite  the 
Table.  Obs.  To  what  are  the  apothecaries'  ounce  and  pound  equal  f 
How  are  drugs  and  medicines  bought  and  sold  ?  151.  In  what  i* 
Long  Measure  used  ?  Recite  the  Table. 


ARTS.  150-152.]  NUMBERS.  141 

OBS.  1.  The  standard  untt  of  Length  adopted  by  the  General  Go- 
"ernment,  is  the  Yard  of  3  feet,  or  36  inches,  and  is  identical  with  the 
imperial  Yard  of  England.  It  is  made  of  brass,  and  is  determined 
from  the  scale  of  Troughton*  at  the  temperature  of  62°  Fahrenheit. 

2.  Long  measure  is  frequently  called  linear,  or  lineal  measure. 
Formerly  the  inch  was  divided  into  3  barleycorns ;  but  the  barleycorn 
is  not  employed  as  a  measure  at  the  present  day.  The  inch  is  com- 
monly divided  either  into  eighths  or  tentlis;  sometimes,  however,  it  ia 
divided  into  twelfths,  which  are  called  lines. 

19.  In  3  feet,  how  many  inches?     In  3  feet  and  4  in., 
how  many  inches  ?     In  4  feet  and  7  in.  ? 

20.  How  many  furlongs  in  3  miles  and  2  furlongs  ? 
How  many  in  4  m.  and  5  fur.  1     In  6  m.  and  7  fur.  ? 

21.  How  many  yards  in  6  feet  ?     In  1 2  ft.  ?     In  1 6  ft.  ? 
In  23  ft.  ? 

22.  How  many  feet  in  27  inches?     In  36  in.  ?     In  41 
in.?     In  64  in.? 

23.  How  many  yards  in  12  feet  ?    In  17  ft  ?    In  25  ft  ? 
In  30  ft? 

CLOTH  MEASURE. 

152*  Cloth  Measure  is  used  in  measuring  cloth,  lace, 
and  all  kinds  of  goods  which  are  bought  and  sold  by  the 
yard. 

24  inches  (in.)  make  1  nail,  marked  no. 


4  nails, or  9  in. 

4  quarters 

3  quarters,  or  f  of  a  yard 

5  quarters,  or  1 4  yard 

6  quarters,  or  1 1  yards 


1  quarter  of  a  yard,  '        qr. 

1  yard,  "      yd. 

1  Flemish  ell,  «  Fl.  e. 

1  English  ell,  "  E.  e. 

1  French  ell,  "  F.  e. 


OBS.  Cloth  measure  is  a  species  of  long  measure.  The  yard  is  th« 
*ame  in  both.  Cloths,  laces,  &c.,  are  bought  and  sold  by  the  linear 
<ard  without  regard  to  their  width. 

QUEST. — Draw  a  line  an  inch  long  upon  the  black-board.  Draw 
one  a  foot,  and  another  a  yard  long.  How  long  is  your  desk  ?  Your 
ieacher's  table ?  How  wide?  How  long  is  the  school  room?  How 
wide  ?  Obs.  What  is  the  standard  unit  of  Length  adopted  by  Congress  ! 
What  is  Long  Measure  often  called  ?  152.  In  what  is  Cloth  Measure 
used  ?  Repeat  the  Table  ?  Obs.  Of  what  is  cloth  measure  a  species  ? 
What  is  the  kind  of  yard  by  which  cloths,  laces,  &c.  are  bought  and 
sold? 

*  A  celebrated  English  artist 


.42  COMPOUND  [SECT.  VII, 

24.  In  3  quarters,  how  many  nails  ?     In  5  qrs.  ?     In 

7  qrs.  ? 

25.  How  many  quarters  in  4  yards  ?     In  6  yds.  ?     In 

8  yds.  ?     In  1 1  yds.  ?     In  15  yds.  ? 

26.  How  many  quarters  in  5  Flemish  ells  ?  In  7  Fl.  e.  ? 
In  10  Fl.  e.? 

27.  How  many  quarters  in  4  English  ells  ?  In  6  E,  e.  ? 
[n9E.  e.? 

28.  In  10  quarters,  how  many  yards?     In  12  qrs.? 
In  15  qrs.  ?     In  18  qrs.  ?     In  24  qrs.  ?     In  30  qrs.  ? 

29.  How  many  French  ells  in  12  quarters?     In  24 
qrs.  ?     In  36  qrs.  ?     In  45  qrs.  ? 


SQUARE  MEASURE. 

153*  Square  Measure  is  used  in  measuring  surfaces 
or  things  whose  length  and  breadth  are  considered  with 
out  regard  to  heighth  or  depth  ;  as,  land,  flooring,  plas 
tering,  &c. 

144  square  inches  (<g.  in.)    make  1  square  foot,  marked  sq.ft 

9  square  feet  "     1  square  yard,  sq.  yd 


40  square  rods  "     1  rood, 

4  roods,  or  160  square  rods  "      1  acre, 
640  acres  "     1  square  mile, 


OBS.  1.  A  square  is  a  figure  which  has 
four  equal  sides,  and  all  its  angles  right  an- 
gles, as  seen  in  the  first  diagram.  Hence, 

A  Square  Inch  is  a  square,  whose  sides 
are  each  a  linear  inch  in  length. 

A  Square  foot  is  a  square,  whose  sides 
we  each  a  linear  foot  in  length. 


QUEST.— 153.  In  what  is  Square  Measure  used  ?  Recite  the  Table. 
Obs.  What  is  a  square  ?  Draw  a  square  upon  the  black-board.  Whal 
Vs  a  square  inch  ?  A  square  foot  ? 


ARTS.  153,  154.] 


NUMBERS. 


A  Square  Yard  is  a  square,  whose  sides 
are  each  a  linear  yard  or  three  linear  feet  in 
length,  and  contains  9  square  feet,  as  repre- 
sented in  the  adjacent  figure. 

2.  In  measuring  land,  surveyors  use  a 
chain  which  is  4  rods  long,  and  is  divided  into 
100  links.  Hence,  25  links  make  1  .rod,  and 
7-r*[$r  inches  make  1  link. 

This  chain  is  commonly  called  Chinter'a 
Chain,  from  the  name  of  its  inventor. 


142 

9  sq.  Jl.  =1  sq.  yd. 

30.  How  many  inches  in  2  square  feet  ? 

31.  How  many  square  feet  in  2  square  yards?     In  3 
yds.  ?     In  5  yds.  ?     In  8  yds.  ? 

32.  How  many  square  yards  in  2  square  rods? 

33.  How  many  roods  in  80  square  rods  ? 

34.  How  many  acres  in  16  roods?     In  25  R?     In 
30  R.?     In48R.? 


CUBIC  MEASURE. 


1  5  4.  Cubic  Measure  is  used  in  measuring  solid  bodies, 
or  things  which  have  length,  breadth,  and  thickness,  such 
as  timber,  stone,  boxes  of  goods,  the  capacity  of  rooms, 
ships,  &c. 


1728  cubic  inches  (cu.  in.)  make  1  cubic  foot, 
27  cubic  feet 
40  feet  of  round,  or 
50  ft.  of  hewn  timber 
42  cubic  feet 


16  cubic  feet 

8  cord  feet,  ot 
128  cubic  feet 


1  cubic  yard, 
1  ton,  or  load, 

1  ton  of  shipping, 

1  foot  of  wood,  or 

a  cord  foot, 

1  cord, 


marked  cu.ft. 
"        cu.  yd. 

M  '7"' 


C. 


OBS.  1.  A  pile  of  wood  8  feet  long,  4  feet  wide,  and  4  feet  high 
contains  1  cord.    For,  8X4X4=128. 


QUEST.  —  What  is  a  square  yard  ?    Draw 

154.  In  what  is  Cubic  Measure  used  ? 


foot;  a  square  yard. 
the  Table. 


square  inch  ;  a  square* 
Recite 


144 


COMPOUND. 


[SECT.  Vli 


2.  A  Cube  la  a  solid  body  bound-  Cubic  Inch. 
ed  by  six  equal  squares.    It  is  often 

called  a  hexaedron.     Hence, 

A  Cubic  Inch  is  a  cube,  each  of 
whose  sides  is  a  square  inch,  as  re- 
presented by  the  adjoining  figure. 

A  Cubic  Foot  is  a  cube,  each  of 
whose  sides  is  a  square  foot. 

3.  The  Cubic  Ton  is  chiefly  used  for 
estimating  the  cartage  and  transpor- 
tation of  timber.     By  a  ton  of  round 
timber  is  meant,  such  a  quantity  of 
timber  in  its  rough  or  natural  state, 

as  when  hewn,  will  make  40  cubic  feet,  and  is  supposed  to  be  equal 
in  weight  to  50  feet  of  hewn  timber. 

Note. — For  an  easy  method  of  forming  models  of  the  cube  and 
other  regular  solids,  see  Thomson's  Legendre's  Geometry,  p.  222. 

WINE  MEASURE. 

155*  Wine  Measure  is  used  in  measuring  wine,  al 
cohol,  molasses,  oil,  and  all  other  liquids  except  beer,  ale, 
and  milk. 

4  gills  (gi.) 

2  pints 

4  quarts 
3 1£  gallons 
42  gallons 
63  gallons,  or  2  barrels 

2  hogsheads  "        pipe  or  butt, " 

2  pipes  «        tun,  "  tun 

OBS.  The  standard  unit  of  LAquid  Measure  adopted  by  the  Govern 
ment,  is  the  English  Wine  Gallon  of  231  cubic  inches,  equal  t*. 
8 1  o  (Hi  pounds  avoirdupois  of  distilled  water  at  the  maximum  density , 
which  is  about  40°  Fahrenheit.* 


make  1  pint,         marked        pi 

a 

quart, 

qt 

u 

gallon, 

gal 

u 

barrel, 

"  bar.  or  bbl 

ft 

tierce, 

"            tier 

it 

hogshead, 

"            hhd 

QUEST. — Obs.  What  is  a  cube?  What  is  a  cubic  inch?  A  cubic 
foot?  Draw  a  cubic  inch  upon  the  black-board.  What  is  meant  by  a 
ion  of  round  timber ?  155.  In  what  is  Wine  Measure  used?  Recite 
the  Table.  Obs.  What  is  the  standard  unit  of  Liquid  Measure  of  the 
United  States  ?  How  many  cubic  inches  in  a  wine  gallon  ? 

*  CHmsted's   Philosophy.    Also,  Hassler  on  Wrights  and  Measures,  p.  102, 


ARTS.  155-157.]  NUMBERS.  145 

35.  In  4  pints,  how  many  gills  ?     In  6  pts.  ?     In   12 
pts.? 

36.  In  5  quarts,  how  many  pints  ?     In  6  qts.  ?     In  9 
qts.?     In  13  qts.? 

37.  In  4  gallons,  how  many  quarts  ?     In  6  gals.  ?     In 
10  gals.?     In  12  gals.? 

38.  In  3  hogsheads,  how  many  barrels  ?     In  7  hhds.  ? 

39.  How  many  quarts  iti  8  pints  ?     In  1 1  pts.  ?     In 
15  pts.?     In  18  pts.? 

40.  How  many  gallons  in    16  quarts?     In  22  qts.? 
In  32  qts.  ? 

BEER  MEASURE. 

156.  Beer  Measure  is  used  in  measuring  beer,  ale, 
and  milk. 

2  pints  (pts.)          make  1  quart,       marked  qt. 

4  quarts  "  1  gallon,  gal 

36  gallons  "  1  barrel,  "  bar.  or  bbl 

1-j  barrels  or  54  gals.  "  1  hogshead,  u  hhd. 

OBS.  The  beer  gallon  contains  282  cubic  inches.     In  many  places 
milk  is  measured  by  wine  measure. 

41.  In  3  quarts,  how  many  pints  ?     In  7  qts.  ?     In  15 
qts.  ? 

42.  In  4  gallons,  how  many  quarts  ?     In  6  gallons  ? 
In  20? 

DRY  MEASURE. 

157.  Dry  Measure  is  used  in  measuring  grain,  fruit, 
Kilt,  &c. 

2  pints  (pt.)           make  1  quart,  marked  qt. 

8  quarts                     "  1  peck,  pk 

4  pecks  or  32«qts.      "  1  bushel,             "  bu. 

8  bushels                   "  1  quarter            "  qr. 

32  bushels                  "  1  chaldron,         u  ch. 

OBS.  1.  In  England,  36  bushels  of  coal  make  a  chaldron. 


QUEST.— 156.  In  what  is  Beer  Measure  used  ?    Recite  tha  Table. 
157.  In  what  is  Dry  Measure  used  ?     Repeat  tke  Tablo. 


146  COMPOUND  [SECT  VIL 

2.  The  standard  unit  of  Dry  Measure  adopted  by  the  Govern- 
ment, is  the  Winchester  Bushel  of  2150-^  cubic  inches,  equal  to 
TVI^TA  pounds  avoirdupois  of  distilled  water,  at  the  maximum  den- 
sity. The  Winchester  bushel  is  so  called,  because  the  standard 
measure  was  formerly  kept  at  Winchester,  England. 

43.  In  4  quarts,  how  many  pints  ?     In  6  qts.  ?     In  10 
qts.  ?     In  14  qts.  ?     In  18  qts.  ? 

44.  How  many  quarts  are  there  in   3  pecks  ?     In  5 
pecks  ? 

45.  How  many  pecks  in   3  bushels  ?     In  5  bu.  ?     In 
10  bu.  ? 

46.  How  many  quarts  in  6  pints?     In    10  pts.  ?     In 

15  pM 

47.  How  many  bushels  in  8  pecks  ?     In  16  pks.  ?     In 
20  pks.  ? 

TIME. 

158*  Time  is  naturally  divided  into  days  and  years ; 
the  "former  are  caused  by  the  revolution  of  the  Earth  on 
its  axis,  the  latter  by  its  revolution  round  the  sun. 

60  seconds  (sec.)  make  1  minute,  marked  min, 

60  minutes                                "      1  hour,  "          hr. 

24  hours  "  1  day,  «  d. 

7  days  1  week,  "  wk 

4  weeks  "  1  lunar  month,  "  mo. 

12  calendar  months,  or  \  « 


365  days  and  6  hrs.,  (nearly) 


1  civil  year,  "         yr. 


OBS.  1.  A  Solar  year  is  the  exact  time  in  which  the  earth  revolves 
round  the  sun,  and  contains  365  days,  5  hours,  48  minutes,  and  48 
seconds. 

2.  Since  the  civil  year  contains  365  days  and  6  hours,  (nearly,)  it  is 
plain  that  in  four  years  a  whole  day  will  be  Drained,  and  therefore 
every  fourth  year  must  have  366  days.  This  is  called  Bissextile,  or 
Leap  Year.  The  odd  day  is  added  to  the  month  of  February ;  in 
every  Leap  year,  therefore,  February  has  29  days. 


QUEST. — Obs.  What  is  the  standard  unit  of  Dry  Measure  adopted  by 
the  government  ?  158.  How  is  time  naturally  divided  ?  Recite  the 
Table.  Obs.  What  is  a  solar  year  ?  How  is  leap  year  occasioned  / 
To  which  month  is  the  odd  day  added  ? 


A**s.  158,  159.] 


NUMBERS. 


147 


3.  The  following  are  the  names  of  the  12  calendar  months  into 
which  the  civil  year  is  divided,  with  the  number  of  days  in  each; 


January, 

February, 

March, 

April, 

May, 

June, 

July, 

August, 

September, 

October, 

November, 

December, 


Jan.) 
Feb.) 
Mar.) 

first 
second 
third 

Apr.) 
May). 

fourth 
jfih 

June) 

sixth 

July) 

seventh 

Aug.) 

eighth 

Sept.) 

ninth 

Oct.) 

tenth 

Nov.) 

eleventh 

Dec.) 

twelfth 

month,  has  31  days. 
•    28    « 
«    31    " 

30 

31 

30 

31 

31 

30 

31 

30 

31 


The  number  of  days  in  each  month  may  be  easily  remembered 
from  the  followiAg  lines : 

"Thirty  days  hath  September, 
April,  June,  and  November; 
February  twenty-eight  alone, 
All  the  rest  have  thirty-one  j 
Except  in  Leap  year,  then  is  the  time, 
When  February  has  twenty-nine." 

48.  How  many  days  in  3  weeks  ?     In  4  wks.  ?     In  5 
vvks.  ?     In  7  wks.  ?     In  9  wks.  ? 

49.  How  many  weeks  in  14  days?     In  21  days?     In 
,  32  days  ?     In  35  days  ?     In  40  days  ? 


CIRCULAR  MEASURE. 


159*  Circular  Measure  is  applied  to  the  divisions  of 
the  circle,  and  is  used  in  reckoning  latitude  and  longi- 
tude, and  the  motion  of  the  heavenly  bodies. 


60  seconds  (")  make  1  minute,  marked  ' 

60  minutes  "  1  degree,         «       ° 

30  degrees  "  1  sign,  "       *.'• 

12  signs,  or  360°  «  1  circle,          "       c. 


QUEST.-— 159.  In  what  is  Circular  Measure  used  ?  Repeat  the  Table, 


148 


COMPOUND 


[SECT.  VU 


OBS.  1.  The  circumference  of 
every  circle  is  divided,  or  supposed 
to  be  divided,  into  360  equal  parts, 
called  degrees,  as  in  the  subjoined 
figure. 

2.  Since  a  degree  is  TeT  part  of 
the  circumference  of  a  circle,  it 
b  obvious  that  its  length  must  de- 
pend on  the  size  of  the  circle. 


270" 


50.  In  2  degrees,  how  many  minutes  ?     In  3  degrees  ? 

51.  In  2  signs,  how  many  degrees  ?     In  3  signs,  how 
many  ?     In  4  signs,  how  many  ? 

«2.  How  many  signs  in  60  degrees?     In  90  degrees? 

MISCELLANEOUS    TABLE. 


12  units 

12  dozen,  or  144 
12  gross,  or  1728 
20  units 
56  pounds 
100  pounds 
30  gallons 

200  Ibs.  of  shad  or  salmon 
196  pounds 
200  pounds 

14  pounds  of  iron,  or  lead 
21  i  stone 
8  pigs 

24  sheets  of  paper 
20  quires 

A  sheet  folded  in  two  leaves,  is  called  a.  folio. 
A  sheet  folded  in  four  leaves,  is  called  a  quarto,  or  4fo. 
A  sheet  folded  in  eight  leaves,  is  called  an  octavo,  or  8vo. 
A  sheet  folded  in  twelve  leaves,  is  called  a  duodecimo,  cr  12?7U), 
A  sheet  folded  in  eighteen  leaves,  is  called  an  I8mo. 

OBS.  Formerly  112  pounds  were  allowed  fora  quintal. 


make  1  dozen,  (doz.) 

"     1  gross. 

"    1  great  gross. 

"     1  score. 

«     1  firkin  of  butter. 

"     1  quintal  of  fish. 

"     1  bar.  of  fish  in  Mass. 

1  bar.  in  N.  Y.  and  Conn. 

1  bar.  of  flour. 

1  bar.  of  pork. 

1  stone. 

1  P^ 

1  fother. 

1  quire. 

1  ream. 


QUEST. — Obs.  How  is  the  circumference  of  every  circle  divided  1 
On  what  does  the  length  of  a  degree  depend  ? 


ART.  160.]  NUMBERS.  149 

REDUCTION  OF  COMPOUND  NUMBERS. 

16O»  The  process  of  changing  compound  numbers 
from  one  denomination  into  another,  without  altering 
their  value,  is  called  REDUCTION. 

EXEECISES   FOR   THE   SLATE. 

Ex.  1.  Reduce  £3  to  farthings. 

Operation.  We  first  reduce  the  given  pounds 

£3  to  shillings.     This  is  done  by  mul- 

20s  in  1  £      ^plying"  them  by  20,  because  20s. 

'  make  £1.      (Art.   147.)     That  is, 

60  shillings,     since  there  are  20s.  in  £1,  in  £3 
12d.  in  Is.        there  are  3  times  20s.  or  60s.     We 
now  reduce  the  60s.  to  pence,  by 
720  pence.          multiplying  them  by   12,  because 
4  far.  in  Id.     12d.   make    Is.      Finally,    we  re- 
duce the   720d.    to  farthings,    by 
ATM.  2880  far.  multiplying   them   by   4,    because 

4  far.  make  Id.     The  last  product,  2880  far.,  is  the  an- 
swer ;  that  is,  £3=2880  far. 

2.  Reduce  £2, 3s.  6d.  and  2  far.  to  farthings. 

Operation.  In  this  example  there  are  shil- 

P  d     f          Im£s5  Pence>  and  farthings.    Hence. 

2      3      6      2         wnen  tne   pounds  are  reduced  to 

20s   in  £1  shillings,   the   given   shillings   (3) 

must  be  added  mentally  to  the  pro- 

43  shillino-s.  duct.     In    like  manner  when  the 

12d.  in  Is.  shillings  are  reduced  to  pence,  the 

given  pence   (6)   must  be  added ; 

522  pence.  and  when  the  pence  are  reduced  to 

4  *ar-  m  W?  farthings,   the  given  farthings  (2) 

2090  far.   Ans.  must  be  added. 

OBS.  In  these  examples  it  is  required  to  reduce  higher  denomina- 
tions to  lower;  as  pounds  to  shillings,  shillings  to  pence,  &c.  This 
»s  done  by  successive  multiplications. 

QUEST. — 160.  What  is  Reduction  ?  How  are  pounds  reduced  to 
ihillingi  1  Why  multiply  by  20  ?  How  are  shillings  reduced  to  pence? 
Why  ?  How,  pence  to  farthings ?  Why  ? 


150  REDUCTION.  [SECT.  VII. 

I6O.  a.  It  often  happens  that  we  wish  to  reduce 
'.ower  denominations  to  higher,  as  farthings  to  pence, 
pence  to  shillings,  and  shillings  to  pounds.  Thus, 

3.  In  2880  farthings,  how  many  pounds  ? 

First,  we  reduce  the  given  farthings 
Operation.       to  pence?  which  is  the  next  higher  de- 

4)2880  far.       nomination.     This  is  done  by  dividing 

l9V79ffH  tnem  by"  4.     For,  since  4  far.  make  Id., 

(Art.   147,)   in   2880  far.  there  are  as 

20)60s.  many  pence,  as  4  is  contained  times  in 

£3  Ans.     2880  ;  and  4  is  contained  in  2880,  720 

times.     We  now  reduce  the  720  pence 

to  shillings,  by  dividing  them  by  12,  because  12d.  make 

Is.     Finally,  we  reduce  the  shillings  (60)  to  pounds,  by 

dividing  by  20,  because  20s.  make  £1.     Thus,  2880  far. 

=£3,  which  is  the  answer  required. 

4.  How  many  pounds  in  2090  farthings? 

Operatien.  In   dividing  by  4  there   is  a 

4)2090  fa,r.  remainder  of  2  far.  ;  in  dividing 


12)522d:2far.over.  bY  ^V  fhere  is  a  remainder  of 
6d.  ;  in  dividing  by  20,  the  quo- 
2°)43s-  6d.  over.      tient  is  £2  and  3s    over.     The 

£2,  3s,  over.      answer,  therefore,  is  £2,  3s.  6d. 
Ans.  £2,  3s.  6d.  2  far.  2   far.     That  is,  2090  far.=£2, 
3s.  6d.  2  far. 

OBS.  1.  The  last  two  examples  are  exactly  the  reverse  of  the  first 
two  ;  that  is,  lower  denominations  are  required  to  be  reduced  to  high- 
er, which  is  done  by  successive  divisions. 

2.  Reducing  compound  numbers  to  lower  denominations  is  usually 
called  Reduction  Descending;  reducing  them  to  higher  denomina- 
tions, Reduction  Ascending.  The  former  employs  multiplication  ;  the 
latter  division.  They  mutually  prove  each  other. 


QUEST. — Ex.  3.  How  are  farthings  reduced  to  pence  ?  Why  divide 
by  4  ?  How  reduce  pence  to  shillings  1  Why  ?  How  shillings  to 
pounds  ?  Why  ?  Obs.  What  is  reducing  compound  numbers  to  lower 
denominations  usually  called  ?  To  higher  denominations  ?  Which  of 
the  fundamental  rules  is  employed  by  the  former  ?  Which  by  the 
latter  1 


ARTS.  161  162.)          REDUCTION.  151 

161*  From  the  preceding  illustrations  we  derive 
the  following 

GENERAL  RULE  FOR  REDUCTION. 

t.  To  reduce  compound  Nos.  to  lower  denominations. 

Multiply  tJie  highest  denomination  given,  by  that  number 
which  it  takes  of  the  next  lower  denomination  to  make  ONE  of 
this  higher ;  to  the  product,  add  the  number  expressed  in  this 
lower  denomination  in  the  given  example.  Proceed  in  this 
manner  with  each  successive  denomination,  till  you  come  to 
the  one  required. 

II.  To  reduce  compound  Nos.  to  higher  denominations. 

Divide  the  given  denomination  by  that  number  which  it 
takes  of  this  denomination  to  make  ONE  of  the  next  higher. 
Proceed  in  this  manner  with  each  successive  denomination, 
till  you  come  to  the  one  required:  The  last  quotient,  with  th& 
several  remainders,  will  be  the  answer  sought. 

162.  PROOF. — Reverse  the  operation;  that  is,  reduce, 
back  the  answer  to  the  original  denominations,  and  if  the 
result  correspond  with  the  numbers  given,  the  work  is  right. 

OBS.  Each  remainder  is  of  the  same  denomination  as  the  dividend 
from  which  it  arose.  (Art.  66.  Obs.  2.) 

STERLING  MONEY.    (ART.  147.) 
5.  In  £35,  4s.  6d.  how  many  pence  ? 

Operation.  Proof. 

£    s.     d.  12)8454  pence. 


35     4     6 

20 

20)704s.  6d. 
£35,  4s.  6d. 

704 
12 

AJU.    8454d. 

QUEST. — 161.  How  are  compound  numbers  reduced  to  lower  denom- 
inations ?  How  reduced  to  higher  denominations  ?  162.  How  is  Re- 
duction proved  ?  Obi.  Of  what  denomination  is  each  remainder  \ 


152  REDUCTION.  [SECT.  VIL 

6.  In  57600  farthings,  how  many  pounds? 

Operation.  Proof. 

4)57600  far.  £60 

12)14400  d.  20 

20)  1200s.  1200  s. 

£60  Ans.  12 

14400  d. 
4 


57600  far. 

7.  In  £43,  12s.,  how  many  shillings  ? 

8.  In  1 7  shillings,  how  many  farthings  ? 

9.  In  1 1 76  pence,  how  many  pounds  ? 

10.  In  12356  farthings,  how  many  shillings? 

11.  In  175  pounds,  how  many  farthings? 

12.  In  £84,  16s.  7-£d.,  how  many  farthings  ? 

13.  In  25256  pence,  how  many  pounds? 

14.  In  56237  farthings,  how  many  pounds? 

15.  In  £25,  9s.  7-£d.,  how  many  farthings? 

TROY  WEIGHT.    (ART.  148.) 

16.  In  11  Ibs.,  how  many  pennyweights? 

Ans.  2640  pwts. 

17.  In  15  ounces,  how  many  grains  ? 

18.  In  10  Ibs.  5  oz.  6  pwts.,  how  many  grains  ? 

19.  In  512  pwts.,  how  many  pounds? 

20.  In  2156  grains,  how  many  ounces? 

21.  In  35210  grains,  how  many  pounds? 

AVOIRDUPOIS  WEIGHT.     (ART.  149.) 

22.  Reduce  25  pounds  to  dran^     Ans.  6400  drams. 

23.  Reduce  36  cwt.  2  qrs.  to  pounds. 

24.  Reduce  5  tons,  7  cwt.  15  Ibs.  to  ounces. 

25.  Reduce  3  quarters,  15  Ibs.  10  oz.  to  drams. 

26.  Reduce  875  ounces  to  pounds. 

27.  Reduce  1565  pounds  to  hundred  weight 
28   Reduce  1728  drams  to  pounds. 


ART.  162.J  REDUCTION.  153 

29.  Reduce  5672  ounces  to  hundred  weight. 

30.  Reduce  15285  pounds  to  tons. 

31.  Reduce  26720  drams  to  hundred  weight. 

APOTHECARIES'  WEIGHT.  (ART.  150.) 

32.  How  many  drams  are  there  in  70  pounds  ? 

Ans.  6720  drams. 

33.  How  many  scruples  in  156  pounds  ? 

34.  How  many  ounces  in  726  scruples  ? 

35.  How  many  pounds  in  1260  drams  ? 

LONG  MEASURE.  (ART.  151.) 

36.  In  96  rods,  how  many  feet  ? 

2)96 

5^-  yds.  in  1  r.  We  .first  multiply  by  5,  then 

"^QQ  by  i,  and  unite  the  two  results. 

43  (Art.  134.  a.)     But  to  multi- 

•=—        ,  ply  by  i,  we  take  half  of  the 

3  ifnfi    d  multiplicand  once.  (Art.  132.) 

Ans.  1584  feet. 

37.  In  45  furlongs,  how  many  inches  ? 

38.  In  1584  feet,  how  many  rods? 

3)1584  We  first  reduce  the  feet  to  yards, 

528  fry  dividing  by  3  ;  next,  reduce  the 

2  yards  to  rods,  by  dividing  by  5£. 

Divide  %  ^  2-5  we  reduce  it  to 


1  T  \inqfi  2-5 

)1U5b  halves,  and  also  reduce  the  dividend 

Ans.  96  (528  yds.)  to  halves,   then  divide 

1056  by  11.  (Art.  139,1.) 

39.  In  1728  inches,  how  many  rods? 

40.  In  26400  feet,  how  many  miles  ? 

41.  In  25  leagues,  how  many  inches? 

42.  In  40  leagues,  6  furlongs,  2  in.,  how  many  inches? 

43.  In  750324  inches,  how  many  miles  ? 

44.  How  many  inches  in  the  circumference  of  the 
«arth  1 


154  REDUCTION.  [SECT.  VII 

CLOTH  MEASURE.  (ART.  152.) 

45.  How  many  quarters  in  45  yards  ? 

46.  How  many  nails  in  53  Flemish  ells  ? 

47.  How  many  nails  in  8 1  English  ells  ? 

48.  Reduce  563  quarters  to  yards. 

49.  Reduce  1824  nails  to  French  ells. 

50.  Reduce  5208  nails  to  English  ells. 

SQUARE  MEASURE.  (ART.  153.) 

51.  In  1766  square  rods  and  19  yards,  how  many  feet  1 

52.  In  56  acres  and  3  roods,  how  many  square  teet? 

53.  In  1275  square  miles,  how  many  acres? 

54.  How  many  square  rods  in  25640  feet  ? 

55.  How  many  acres  in  1865  roods? 

56.  How  many  acres  in  2118165^  yards'? 

1G3.  The  area  of  a  floor,  .a  piece  of  land,  or  any 
surface  which  has  four  sides  and  four  right  angles,  is  found 
by  multiplying  its  length  and,  breadth  together. 

Note. — The  area  of  a  figure  is  the  superficial  contents  or  space  con- 
tained within  the  line  or  lines,  by  which  the  figure  is  bounded.  It  ia 
reckoned  in  square  inches,  feet,  yards,  rods,  &c. 

57.  How  many  square  feet  are  there  in  a  table  which 
is  4  feet  long  and  3  feet  wide? 


Suggestion. — Let  the  given 
table  be  represented  by  the 
subjoined  figure,  the  length  of 
which  is  divided  into  4  equal 
parts,  and  the  breadth  into  3 
equal  parts,  which  we  will  call 
linear  feet.  Now  it  is  plain 
that  the  table  will  contain  as 
many  sq.  feet  as  there  are  squares  in  the  given  figure.  But 


QUEST. — 163.  How  do  you  find  the  area  or  superficial  contents  of  a 
surface  having  four  sides  and  four  right  angles  ?  Note.  What  is  meant 
by  the  terra  area  ?  How  is  it  reckoned  ?  Obs.  What  is  a  figure  which 
nas  four  siJss  and  four  right  angles,  called  ? 


ARTS.  163,  164.]  REDUCTION.  155 

the  number  of  squares  in  the  figure  is  equal  to  the  number 
of  equal  parts  (linear  feet)  which  its  length  contains,  re- 
peated as  many  times  as  ihere  are  equal  parts  (lineai 
feet)  in  its  breadth;  that  is,  equal  to  4x3,  or  12.  The 
fable  therefore  contains  12  square  feet. 

OBS.  A  figure  which  has  four  sides  and  four  right  angles,  like  the 
preceding,  is  called  a  Rectangle,  or  Parallelogram. 

58.  What  is  the  area  of  a  garden,  which  is  8  rods  long 
and  5  rods  wide  ?  Ans.  40  square  rods. 

59.  How  many  square  feet  in  a  floor,  18  feet  Ions:  and 
17  feet  wide? 

60.  How  many  square  yards  in  a  ceiling,  20  feet  long 
and  18  feet  wide? 

61.  What  is  the  area  of  a  field,  which  is  36  rods  long 
and  25  rods  wide  ? 

62.  How  many  acres  are  there  in  a  piece  of  land,  80 
rods  long  and  48  rods  wide  ? 

CUBIC  MEASURE.  (ART.  154.) 

63.  In  75  cubic  feet,  how  many  inches  ? 

64.  In  37  tons  of  round  timber,  how  many  inches  ? 

65.  In  28124   cubic   feet,   how  many  tons  of  hewn 
timber  ? 

66.  In  16568  cubic  feet  of  wood,  how  many  cords? 

67.  In  65  cords  of  wood,  how  many  cubic  feet  ? 

164.  The  solidity ;  or  cubical  contents  of  boxes  ol 
goods,  piles  of  wood,  &c.,  are  found  by  multiplying  the 
length,  breadth,  and  thickness  together. 

68.  How  many  cubic  inches  are  there  in  a  box,  wnose 
length  is  30  inches,  its  breadth  18,  and  its  depth  15  inches? 

Ans.  8100  cu.  in. 

69.  How  many  cubic  inches  in  a  block  of  marble,  43 
inches  long,  18  inches  broad,  and  12  inches  thick? 

QUEST. — 164.  How  are  the  cubical  contents  of  a  box  of  goods,  a  pile 
•fwood,  &c.,  found? 


150  COMPOUND  [SECT.  VIL 

70.  How  many  cubic  feet  in  a  room,  16  fe^i  long,  15 
teet  wide,  and  9  feet  high  ? 

71.  How  many  cubic  feet  in  a  load  of  wood,  8  feet  long., 
4  feet  wide,  and  3-£  feet  high  ? 

72.  How  many  cubic  feet  in  a  pile  of  wood,  16  feet 
long,  6  feet  wide,  and  5  feet  high  ?     How  many  cords  ? 

73.  How  many  cords  of  wood  in  a  pile,   140  feet  long 
4  £  feet  wide,  and  6£  feet  high  ? 

WINE  MEASURE.  (ART.  155.) 

74.  In  4624  gills,  how  many  gallons  ? 

75.  In  24260  quarts,  how  many  hogsheads  ? 

76.  How  many  pints  in  1 5  hogsheads,  and  20  gallons  ? 

77.  How  many  gills  in  40  barrels  ? 

BEER  MEASURE.  (ART.  156.) 

78.  How  many  barrels  of  beer  in  5000  pints  ? 

79.  How  many  hogsheads  in  7800  quarts  ? 

80.  How  many  quarts  in  25  hogsheads,  and  7  gallons  1 

81.  How  many  pints  in  110  gallons,  3  qts.  and  I  pt.  1 

DRY  MEASURE.    (ART.  157.) 

82.  Reduce  536  bushels,  and  3  pecks  to  quarts. 

83.  Reduce  821  chaldrons  to  pints. 

84.  Reduce  1728  pints  to  pecks. 

85.  Reduce  85600  quarts  to  bushels. 

TIME.  (ART.  153.) 

86.  In  1 5  days,  6  hours,  and  9  min.,  how  many  seconds  V 

87.  In  365  days  and  6  hours,  how  many  minutes  ? 

88.  How  many  seconds  in  a  solar  year  ? 

89.  Allowing  365d.  6h.  to  a  year,  how  many  mmutet 
has  a  person  lived  who  is  21  years  old  ? 

90.  How  many  hours  in  568240  seconds  ? 

91.  How  many  weeks  in  8568456  minutes? 

92.  How  many  lunar  months  in  6925600  hours  ? 

93.  How  many  years  in  56857200  hours? 

94.  How  many  years  in  1 000000000  seconds  ? 


A.RT.    165.]  NUMBERS.  157 

CIRCULAR  MEASURE.   (ART.  159.) 

95.  In  75  degrees,  how  many  seconds  ? 

96.  In  8  signs,  and  15  degrees,  how  many  minutes? 

97.  In  12  signs,  how  many  seconds? 

98.  In  86860  seconds, how  many  degrees  ? 

99.  In  567800  minutes,  how  many  signs  ? 
100.  In  25000000  seconds,  how  many  signs? 

COMPOUND  NUMBERS  REDUCED  TO  FRACTIONS. 
Ex.   1.  Change  7s.  6d.  to  the  fraction  of  a  pound. 

7s.  6d.  £1  or  20s.    7, 

12      (Art.  161,  I,)  we 
Numerator  90d.     Denominator  240d.    have  90d.  which 
Ans.  £2a3Qo=i2ar,  or  £f.  js  me  numerator 

of  the  fraction.     Then  reducing  £1  to  the  same  denomi- 
nation as  the  numerator,  we  have  240d.,  which  is  the 
denominator.     Consequently  •£&  is  the  fraction  required. 
But  -5-W  may  be  reduced  to  lower  terms.     Thus  •£&=* 
ft,oii.     (Art.  120.)     Hence, 

165*  To  reduce  a  compound  number  to  a  common 
fraction  of  a  higher  denomination. 

First  reduce  the  given  compound  number  to  the  lowest 
denomination  mentioned  for  the  numerator;  then  reduce  a 
TOUT  of  the  denomination  of  the  required  fraction  to  the 
same  denomination  as  the  numerator,  and  the  result  will  be 
the  denominator.  (Art.  161.) 

OBS.  When  the  given  number  contains  but  one  denomination,  it  of 
course  requires  no  reduction. 

2.  Reduce  3s.  7d.  2  far.  to  the  fraction  of  £1. 

Ans.  £££fr.  or  £-£&. 

3.  Reduce  9d.  3  far.  to  the  fraction  of  Is. 

4.  What  part  of  a  bushel  is  3  pecks  and  5  qts.  ? 

QUEST. — 165*  How  is  a  compound  number  reduced  to  a  common 


158  COMPOUND  [SECT.  VII 

5.  What  part  of  a  peck  is  5  qts.  and  1  pt.  ? 

6.  What  part  of  a  gallon  is  3  qts.  1  pt.  and  3  gills  1 

7.  What  part  of  1  gallon  is  1  pt   and  1  gill? 

8.  What  part  of  1  hogshead  is  15  gals,  and  3  qts.? 

9.  What  part  of  1  ton  is  5  cwt.  and  2  qrs.  ? 

10.  What  part  of  1  hundred  weight  is  2  qrs.  and  7  Ibs/ 

11.  What  part  of  1  quarter  is  1  Ib.  and  5  oz.  ? 

12.  What  part  of  1  mile  is  45  rods? 

13.  What  part  of  1  mile  is  10  fur.  and  35  rods  ? 

14.  What  part  of  1  league  is  1  m.  1  fur.  and  1  r.? 

15.  What  part  of  1  yard  is  2  qrs.  and  3  nails? 

16.  What  part  of  £1  is  1  penny?  Ans.  frfrr- 

17.  What  part  of  £1  is  -f  of  a  penny? 

Note. — The  lowest  denomination  mentioned  in  this  example,  is 
thirds  of  a  penny.  Hence,  £1  must  be  reduced  to  thirds  of  a  penny 
for  the  denominator,  and  2,  the  given  number  of  thirds  will  be  the 
numerator.  Ans.  £~T%~5i  or  ^TaT- 

18.  What  part  of  £1  is  5$  shillings?  Ans.  £ff. 

19.  What  part  of  1  day  is  2£  hours  ? 

20.  What  part  of  1  day  is  4  h.  and  8-J-  min.  ? 

21.  What  part  of  1  hour  is  3  rhin.  and  40  sec.  ? 

22.  What  part  of  1  hour  is  15f  sec.? 

23.  What  part  of  1  pound  is  -f-  of  an  ounce  ? 

24.  What  part  of  1  ton  is  •£  of  a  pound  ? 

25.  What  part  of  1  hogshead  is  f  of  a  gallon  ? 

26.  What  part  of  1  gallon  is  -f  of  a  gill  ? 

FRACTIONAL  COMPOUND  NUMBERS 

REDUCED    TO  WHOLE    NUMBERS    OF    LOWER   DENOMINATIONS. 

Ex.   1.  Reduce  f  of  £1  to  shillings  and  pence. 

Operation.  Multiply  the  numerator  by  20,  to 

3  eighths  £.          reduce  it  to  shillings,  as  in  reduc 
20  tion.    (Art  161,  I.)    £ix20s.=^s. 

g\50  or  7s.  and  4  remainder.  Again,  mul- 

tiplying- the  remainder  4  by  12,  we 
dul.  7,and4rem.      J/e  |g.    an(J   48^8=6d       The 

quotients,  7s.  and  6d.  are  the  an- 
8)48  swer  required.     Hence. 

Pence  6.  Ans.  7s.  6d. 


S.  166,  167.]  NUMBERS.  159 


166.  To  reduce  fractional  compound  numbers  to 
whole  numbers. 

First  reduce  the  given  numerator  to  the  next  lower  denom- 
ination; (Art.  161,  I;)  then  divide  the  product  by  the  de- 
nominator, and  the  quotient  will  be  an  integer  of  the  next 
lower  denomination.  Proceed  in  like  manner  with  the  re- 
mainder, and  the  several  quotients  will  be  the  whole  numbers 
required. 

2.  Reduce  £  of  £1  to  shillings.  Ans.  12s. 

3.  How  many  shillings  and  pence  in  £^  ? 

4.  How  many  shillings,  &c.,  in  £f  ? 

5.  In  •£•  of  1  week,  how  many  days,  hours, 

6.  In  -f^  of  1  day,  how  many  hours,  minutes, 

7.  Change  f  of  1  league  to  miles,  &c. 

8.  Change  \  of  1  mile  to  furlongs,  &c. 

9.  Reduce  -fa  of  1  hundred  weight  to  quarters, 

10.  In  f  of  1  ton,  how  many  hundred  weight,  &c.  ? 

11.  In  -f  of  1  bushel,  how  many  pecks,  quarts,  &c.  ? 

12.  In  -^g-  of  1  peck,  how  many  quarts,  &c.  ? 

13.  Reduce  -fr  of  £1  to  shillings. 

Suggestion.  —  Since  the  numerator,  when  reduced  to  the 
denomination  required,  cannot  be  divided  by  the  denomi- 
nator, the  division  must  be  represented. 


Note.  —  This,  in  effect,  is  reducing  -^  of  £1  to  the  fraction  of  a 
shilling. 


14.  Reduce  yfj  of  £1  to  pence.  Ans.  -fffd. 

167.  From  the  last  two  examples  it  is  manifest,  that 
a  fraction  of  a  higher  denomination  may  be  changed  to  a 
fraction  of  a  lower  denomination,  by  reducing  the  given 
numerator  to  the  denomination  of  the  required  fraction,  and 
'placing  the  result  over  the  given  denominator. 

QUEST.  —  166.  How  are  fractional  compound  numbers  reduced  to 
whole  ones  ?  167.  How  is  a  fraction  of  a  higher  denomination  clanged 
wo  a  fraction  of  a  lower  denomination  t 


160  COMPOUND  [SECT.  VII 


15.  Reduce  Tar  °f  £1  to  tne  fraction  of  a  shilling. 

Ans.  y^ys. 

16.  Reduce  rlir  °f  1  week  to  the  fraction  of  a  day. 

17.  Change  TT*ST  °f  1  m^e  to  tne  fraction  of  a  rod. 

18.  Change  TOT  of  1  rod  to  the  fraction  of  a  foot. 

19.  Change  \Yli  of  1.  yard  to  the  fraction  of  a  nail. 

20.  Change  1  0  010  0  0  of  1  ton  to  the  fraction  of  a  pound. 

ADDITION  OF  COMPOUND  NUMBERS. 

lat  is  the  sum  of  £4,  9s.  6d.  2  far.  :  £3,  12s.  8d 


1 .  What  is  the  sum  of  £4, ' 
3  far.<ftid  £8,  6s.  9d.  1  far.  ? 


tration.  Having  placed  the  farthings 

£       £      d.    far.  under   farthings,  pence   under 

4  "    ,9  "  6  "  2  pence,  &c.5  we  add  the  column 

3  "  1'2  "  8  "  3  of  farthings  together,  as  in  sim- 

8  "    6  "  9  "   1  pie  addition,  and  find  the  sum 

TO"  9  "  0  "  2  Ans  *s  ^»  which  ]S  equal  to  Id.  and 
2  far.  over.  Set  the  2  far.  un- 
der the  column  of  farthings,  and  carry  the  Id.  to  the  col- 
umn of  pence.  The  sum  of  the  pence  is  24,  which  is 
equal  to  2s.  and  nothing  over.  Place  a  cipher  under  the 
column  of  pence,  and  carry  the  2s.  to  the  column  of  shil- 
lings. The  sum  of  the  shillings  is  29,  which  is  equal  to 
£1  and  9s.  over.  Write  the  9s.  under  the  column  of  shil- 
lings, and  carry  the  £1  to  "the  column  of  pounds.  The 
sum  of  the  pounds  is  16,  the  whole  of  which  is  set  down 
in  the  same  manner,  as  the  left  hand  column  in  simple 
addition.  (Art.  25.)  The  answer  is  £16,  9s.  Od.  2  far. 

168.  Hence,  we  derive  the  following  general 
RULE  FOR  ADDING  COMPOUND  NUMBERS. 

1.  Write  the  numbers  so  that  the  same  denominations  shah 
stand  under  each  other. 


QUEST. — 168.  How  do  you  write  compound  numbers  for  addition  ? 
Which  denomination  do  you  add  first  ?  When  the  gum  of  any  column 
is  found,  what  is  to  be  done  with  it ! 


ART.  168.]  ADDITION.  161 

II.  Beginning  with  the  lowest  denomination,  find  the  sum 
of  each  column  separately,  and  divide  it  by  that  number 
which  it  requires  of  the  column  added,  to  make  ONE  of  the 
next  higher  denomination.     Set  the  remainder  under  the 
column,  and  carry  the  quotient  to  the  next  column. 

III.  Proceed  in  this  manner  with  all  the  other  denomina- 
tions except  the  highest,  whose  entire  sum,  is  set  down  as  in 
simple  addition.    (Art.  29.) 

PR.OOF. — The  proof  is  the  same  as  in  Simple  Addition. 

OBS.  1.  Fractional  compound  numbers  should  be  reduced  to  whole 
numbers  of  lower  denominations,  then  added  as  above.  (Art.  166.) 

2.  The  process  of  adding  numbers  of  different  denominations,  is 
called  Compound  Addition.  It  is  the  same  as  Simple  Addition,  except 
in  the  method  of  carrying  from  one  denomination  to  another. 

2.  What  is  the  sum  of  £|,  -f s.  |d.,  and  £f,  -J-s.  ? 

Ans.  £l,4s.  4d.  3|f. 

3.                               4.  5. 

£      s.    d.  far.          £       s.     d.  £      s.      d. 

6742             10     15     8  21     18     10 

0671              16     11     0  1       6     11 

12     15     6     0              25      18     9  35     12       7 


6.  7.  8. 

Ib.   oz.  pwt.  c/r.  oz.  pwt.  gr.  Ib.  oz.  pwt. 

5     8     16  7  15      12     8  12     6     15 

7     9       6  12  11       6     7  19     0       7 

10     6     15  10  10     13     8  18     16 

21     3       4  5  601  28     3     11 


9.  Add  7  Ibs.  9  oz.  16  pwts.  10  grs. ;  3  Ibs.  10  oz.  8 
pwts.  9  grs. ;  8  Ibs.  3  oz.  1  pwt.  4  grs. 

10.  A  man  bought  a  coach  for  £35,  12s. ;  a  horse  for 
£27,  8s.  lOd. ;   a  harness  for  £7,  16s.  lid.:    what  did 
the  whole  cost  ? 

QUEST. — What  is  done  with  the  last  column  ?  How  prove  the  opera' 
tion  ?  Ob*.  How  add  fractional  compound  numbers  ?  What  is  the  pro- 
cess of  adding  compound  numbers  called  ?  Does  it  differ  from  simple 
addition  ? 

6 


162  COMPOUND  [SECT.  VIL 

1 1.  A  merchant  bought  of  one  dairy-man  5  cwt.  1 1  Ibs. 
6  ounces  of  butter ;  of  another,  3  cwt.  15  Ibs.  9  oz. ;  of 
another,  7  cwt.   6  Ibs.  10  oz. :  how  much  did  he  buy  of 
all? 

12.  A  manufacturer  bought   of  one  man  73  Ibs.  of 
wool;  of  another,  96  Ibs.  6  oz.  ;  of  another,    135  Ibs.  11 
oz.  ;  of  another,  320  Ibs.  9  oz. ;  of  another,  642  Ibs.  3 
oz. :  how  much  wool  did  he  buy  ? 

13.  A  naan  sold  to  one  customer  2  tons,  62  Ibs.  10  oz. 
of  hay ;  to  another,  5  tons,  40  Ibs.  12  oz. ;  to  another,  3 
tons,  75  Ibs.  6  oz. :  how  much  did  he  sell  to  all  1 

14.  A  man  wove   7  yds.  3  qrs.  2  na.  of  cloth  in  one 
day ;  the  next  day,  6  yds.  1  qr.  3  na.  ;   the  next,  8  yds. 
3  qrs.  1  na. ;  the  next,  5  yds.  2  qrs.  3  na. :  how  much  did 
he  weave  in  all  ? 

15.  Bought  several  pieces  of  cotton ;    one  contained 
26  yds.    1   qr.    2  na. ;  another,  30  yds.   2  qrs.  ;  another, 
29-£  yds.  3   na. ;  another,  32-^-  yds.    1  na. :    how  many 
yards  did  they  all  contain  ? 

16.  A  hotel  keeper  bought  at  one  time,  15  bu.  2  pks. 
3  qts.  of  oats ;  at  another,  10  bu.  1  pk.  2  qts. ;  at  another, 
20-£  bu.  6  qts.  ;  at  another,  18-£  bu.  5  qts. :  what  was  the 
amount  of  all  his  purchases  ? 

17.  Bought  4  loads  of  wheat;  the  first  containing  23 
bu.  3  pks.  5  qts.  ;  the  second,  20-^-  bu.  6  qts.  ;   the  third. 
26-J-  bu. ;  the  fourth,  2 If  bu.  7  qts. :  how  many  bushels 
did  they  all  contain  ? 

18.  What  is  the  sum  of  16  m.  3  fur.  16  r.  ?  26  m.  1 
fur.  33  r.  ;    10  m.  8  fur.  22  r.  ;   45  m.  7  fur.  20  r.  ? 

19.  A  merchant  bought  3  casks  of  oil ;  one  held  2 
hhds.  30  gals.  2  qts. ;  another,  3  hhds.  10  gals. ;  another, 
1  hhd.  1 3  gals.  1  qt. :  how  much  did  they  all  hold  ? 

20.  Sold  several  lots  of  wine,  in  the  following  quanti- 
ties; 1  pipe,  1  hhd.  21  gals.  2  qts.  1  pt. ;  2  pipes,  11  gals. 
3  qts.  1  pt. ;  3  hhds.  15  gals.  2  qts. ;  3  pipes,  10  gals.  2 
qts.  1  pt. :  how  much  was  sold  in  all  ? 

21.  A  mason  plastered  one  room  containing  45  square 
yards,  7  ft.  6  in. ;  another,  25  yds.  6  ft.  95  in. ;  another, 
38  yds.  4  ft.  41  in. :  what  was  the  amount  of  plastering  in 
all  the  rooms  ? 


A.RT.   169.]  SUBTRACTION.  163 

22.  Sold  10  A.  35  r.  10  sq.  ft.  of  land  at  one  time;  at 
another,  3  A.  10  r.  15  ft. ;  at  another,  ISA.   16  r.  23  ft. : 
what  was  the  amount  of  land  sold  ? 

23.  A  merchant  received  several  boxes  of  goods  ;  one 
contained  16  cu.  ft.  61  in.  ;  another,  25  ft.  81  in. ;  another 
20  ft.  13  in. ;  another,  38  ft.  72  in. :  how  many  cubic  feet 
and  inches  did  they  all  contain  ? 

24.  One  pile  of  wood  contains  IOC.  38  ft.  39  in. ;  an- 
other, 15  C.  56  ft.  73  in. ;  another,  30  C.    19  ft.  44  in. ; 
another,  17  C.  84  ft.  21  in. :  how  much  do  they  all  con- 
tain? 

SUBTRACTION  OF  COMPOUND  NUMBERS. 
Ex.   1-   From  £15,  7s.  6d.  3  far.,  subtract  £6,  4s.  8d.  2  far. 

Operation.  Having  placed  the  less  num- 

£       s.        d.  far.  ber  under  the  greater,  with  far- 

15  "  7  '      6  "  3  things   under   farthings,    pence 

6  "  4  "     8  "  2  under  pence,  &c.,  we  subtract 

~9  "  2  "  10  "  1  Ans.  ^  far.  from  3*  far.,  and  set  the 
remainder  1  far.  under  the  col- 
umn of  farthings.  But  8d.  cannot  be  taken  from  6d.  ; 
we  therefore  borrow  1  from  the  next  higher  denomina- 
tion, which  is  shillings  ;  and  Is.  or  12d.  added  to  the  6d. 
make  18d.  And  8d.  from  18d.  leaves  lOd.  Since  we 
borrowed,  we  must  carry  1  to  the  next  column,  as  in 
simple  subtraction.  1  added  to  4  makes  5 ;  and  5  from  7, 
leaves  2.  6  from  15  leaves  9.  Ans.  £9,  2s.  lOd.  1  far. 

169.  Hence,  we  derive  the  following  general 
RULE  FOR  SUBTRACTING  COMPOUND  NUMBERS. 

I.  Write  the  less  number  under  the  greater,  so  that  the 
same  denominations  may  stand  under  each  other. 

I[.  Beginning  with  the  lowest  denomination,  subtract  the 
number  in  each  denomination  of  the.  lower  line  from  the  num- 
for  above  it.  and  set  the  remainder  below. 

QUEST. — 169.  How  do  you  write  compound  numbers  for  subtraction? 
Where  begin  to  subtract  ?  When  the  number  in  the  lower  line  is  large  r 
than  that  above  it ,  what  is  to  be  done  ?  How  is  tue  operation  DreWy  ? 


164  COMPOUND  [SECT.  VIL 

III.  When  a  number  in  any  denomination  of  the  lowei 
line  is  larger  than  the  number  above  it,  borrow  one  of  tht 
next  higher  denomination  and  add  it  to  the.  number  in  the 
upper  line.  Subtract  as  before,  and  carry  1  to  the  next  de- 
nomination in  the  lower  line,  as  in  subtraction  of  simple 
numbers.  (Art.  40.) 

PROOF. — The  proof  is  the  same  as  in  Simple  Subtraction. 

OBS.  1.  Fractional  compound  numbers  should  be  reduced  to  wholo 
numbers  of  lower  denominations,  then  subtracted  as  above.  (Art.  1 66.) 

2.  The  process  of  finding  the  difference  between  numbers  of  dif- 
ferent denominations  is  called  Compound  Subtraction.  It  does  not 
differ  from  Simple  Subtraction,  except  in  the  mode  of  borrowing. 

2.  From  £f,  f  s.,  take  £-£,  |s. 

Solution.— £$,  fs.=16s.  9d.,  and  £f,  is.=7s.  8d. 

Ans.  9s.  Id. 

3.  From  £15,  16s.  lOd.  3  far.,  take  £7,  8s.  lid.  1  far. 

4.  From  £56,  7s.  6d.  1  far.,  take  £20,  3s.  lOd.  3  far. 

5.  6. 

From  16T.  lOcwt.  3qrs.  fibs.        125T.  7cwt.  2  qrs.  20lbs. 
Take     8T.     5cwt.  Iqr.    2lbs.          96T.  9cwt.  3  qrs.  12lbs. 

7.  8. 

From  16gals.  3qts.  ]pt.  2gi.  121hhds.  28gals.  Iqt. 

Take     Tgals.  2qts.  Opt.  3gi.  63hlids.  21gals.  3qts. 

9.  Bought  2  silver  pitchers,  one  weighing  2  Ibs.  10  oz. 
10  pwts.  7  grs. ;  the  other,  2  Ibs.  3  oz.  12  pwts.  5  grs.: 
what  is  the  difference  in  their  weight  ? 

10.  A  merchant  had  28  yds.  3  qrs.  2  na.  of  cloth,  and 
sold  15  yds.  1  qr.  3  na. :  kow  much  had  he  left? 

11.  A  lady  bought  2  pieces  of  silk,  one  of  which  con- 
tained 19  yds.  2  qrs.  1  na. ;  the  other,  15  yds.  3  qrs.  3  na. : 
what  is  the  difference  in  the  length  ? 

12.  From  25  m.  7  fur.  8  r.  12  ft.  6  in.,  take  16  m.  6  fuu 
30  r.  4  ft.  8  in. 

QUEST. — Obs.  What  is  the  process  of  subtracting  compound  number* 
called  \  Does  it  differ  from  simple  subtraction  ? 


A.RT.  170.]  SUBTRACTION.  165 

13.  A  man  owning  95  A.  75  r.  67  sq.  Tt.  of  land,  sold  40 
A.  86  r.  29  ft. :  how  much  had  he  left  ? 

14.  A  farmer  having  bought   120  A.   3  R.  28  r.  of 
land,  divided  it  into  two  pastures,  one  of  which  contained 
50  A.  2  R.  35  r. :  how  much  did  the  other  contain  ? 

15.  A  tanner  built  two  cubical  vats,  one  containing 
116  ft.  149  in.,  the  other  245  ft.  73  in. :  what  is  the  differ- 
ence between  them  ? 

1.6.  A  man  having  65  C.  95  ft.  123  in.  of  wood  in  his 
shed,  sold  16  C.  117  ft.  65  in. :  how  much  had  he  left  ? 

17.  From  27  yrs.  8  mos.  3  wks.  4  ds.  13  hrs.  35  min. 
Take  19  yrs.  5  mos.  6  wks.  5  ds.  21  hrs.  20  min. 

18.  What  is  the  time  from  July  4th,  1840,  to  March 
1st,  1845? 

Operation. 

Y       mo       d  March  is  the  3d  month,  and 

'  ff      '  July  the  7th.     Since  4  d.  cannot 

be  taken  from  1  d.,  we  borrow  1 

1840  "  7  " ^  mo.  or  30  d  ;  then  say,  4  from  3 1 

4  "  7  "  27  Ans.      leaves  27.     1  to  carry  to  7  makes 
8,  but  8  from  3  is  impossible ; 

we  therefore  borrow  1  yr.  or  12  mos.,  and  say,  8  from  15 
leaves  7.  1  to  carry  to  0  is  1,  and  1  from  5  leaves  4. 
Hence, 

1 7  O.  To  find  the  time  between  two  dates. 

Write  the  earlier  date  under  the  later,  placing  the  years 
on  the  left,  the  number  of  the  month  next,  and  the  day  of  the 
month  on  the  right,  and  subtract  as  before.  (Art.  169.) 

OBS.  1 .  The  number  of  the  month  is  easily  determined  by  reckon- 
ing from  January,  the  1st  mo.,  Feb.  the  2d,  &c.  (Art.  158.  Obs.  3.) 

'2.  In  finding  the  time  between  two  dates,  and  in  casting  interest', 
30  days  are  considered  a  month,  and  12  months  a  year. 

19.  What  is  the  time  from  Oct.  15th,  1835,  to  March 
10th,  1842? 

20.  The  Independence  of  the  United  States  was  de- 

QUEST.— 170.  How  do  you  find  the  time  between  two  dates  ?  Obs. 
in  finding  time  between  two  dates,  and  in  casting  interest,  how  many 
isys  are  considered  a  month  ?  How  many  months  a  year  ? 


166  COMPOUND.  [SECT.  VII. 

elared  July  4th,  *1776.  How  much  time  had  elapsed  on 
the  25th  of  Aug.  1845? 

21.  A  note  dated  Oct.  2d,  1840,  was  paid  Dec.  25th, 
1843  :  how  long  was  it  from  its  date  to  its  payment  ? 

22.  A  ship  sailed  on  a  whaling  voyage,  Aug.  25th. 
1840,  and  returned  April  15th,  1844  :  how  long  was  her 
voyage  ? 

MULTIPLICATION  OF  COMPOUND  NUMBERS. 

Ex.  1.  What  will  5  yards  of  broadcloth  cost,  at  £2,  3s. 
6d.  3  far.  per  yard  ? 

Suggestion. — If  1  yard  costs  £2,  3s.  6d.  3  far.,  5  yards 
will  cost  5  times  as  much. 

Operation.  Beginning  with  the  low- 

£          s       d     far  est  denomination,  we  say,  5 

*   it     o'  ff     '/  times    3   far.    are    15    far.; 

now  15  far.  are  equal  to  3d. 

£  and  3  far.  over.     Set  the  3 

10  "    17  "  9  "3  Ans.     far.  under  the  denomination 
multiplied,  and  carry  the  3d. 

to  the  next  product.  5  times  6d.  are  30d.  and  3d.  make 
33d.,  equal  to  2s.  and  9d.  Set  the  9d.  under  the  pence, 
and  carry  the  2s.  to  the  next  product.  5  times  3s.  are  15s. 
and  2s.  make  17s.  As  the  product  17s.  does  not  make 
one  in  the  next  denomination,  we  set  it  under  the  column 
multiplied.  Finally,  5  times  £2  are  £10.  The  answei 
is  £10,  17s.  9d.  3  far. 

171.  Hence,  we  deduce  the  following  general 
RULE  FOR  MULTIPLYING  COMPOUND  NUMBERS. 

Multiply  each  denomination  separately,  beginning  with 
the  lowest,  and  divide  each  product  by  that  number  which  u 
takes  of  the  denomination  multiplied,  to  make  one  of  the 


QUEST. — 171.  Where  do  you  begin  to  multiply  a  compound  number  ? 
What  is  done  with  each  product  1  Obs.  When  the  multiplier  is  a 
composite  number,  how  proceed  ?  What  is  the  process  of  multiplying 
different  denominations,  called  ? 


ART.   171.]  MULTIPLICATION.  167 

next  higlier ;  set  down  the  remainder,  and  carry  the  quo- 
dent  to  the  7iext  product,  as  in  addition  of  compound  num- 
bers. (Art.  168.) 

OBS.  1 .  When  the  multiplier  is  a  composite  number,  it  is  advisable 
to  multiply  first  by  one  factor  and  that  product  by  the  other.  (Art.  57.) 

2.  The  process  of  multiplying  different  denominations  is  called 
Compound  Multiplication. 

2.  Multiply,  £5,  7s.  Sd.  2  far.  by  18. 

Operation.    • 
£         s.       d.    far. 

5  "     7  "  8  "  2  Multiply  by  the   factors  of 

G  18,  which  are  6  and  3. 

32"^ 6  "  3  "  0~~ 
3 


96  "   18  ''  9  "  0  A?is. 

3.  What  will  5  horses  cost,  at  £25, 10s.  6d.  apiece  ? 

4.  A  company  of  6  persons  agreed  to  pay  £31,  5s.  8d 
apiece  for  their  passage  from  Hamburg  to  New  York  : 
what  was  the  expense  of  their  passage  ? 

5.  What  cost  9  yards  of  cloth,  at  18s.  9fd.  per  yard? 

6.  What  cost  6  pipes  of  wine,  at  £9,  7s.  8-^-d.  apiece  ? 

7.  What  cost  8  cows,  at  £5,  10s.  6d.  apiece  ? 

8.  In  a  solar  year  there  are  365  days,  5  hrs.  48  min. 
48  sec. :  how  many  days,  hours,  &c.,  has  a  person  lived 
who  is  2 1  years  old  ? 

9.  Bought  10  silver  cups,  each  weighing  3  oz.  15  pwts. 
10  grs. :  what  is  the  weight  of  the  whole  ? 

10.  What   is   the  weight   of  72   silver  dollars,   each 
weighing  17  pwts.  8  grs.? 

11.  Bought  7  loads  of  hay,  each  weighing  1  T.  3  cwt. 
3  qrs.  12  Ibs. :  what  is  the  weight  of  the  whole  ? 

12.  What  is  the  weight  of  20  hogsheads  of  molasses, 
each  weighing  5  cwt.  3  qrs.  17  Ibs.  10  oz.? 

13.  A  man  bought  9  oxen,  weighing  1123  Ibs.  15  oz 
apiece :  what  was  the  weight  of  the  whole  ? 

14.  A  grocer  bought  11  casks  of  brandy,  each  contain- 
ing 54  gals.  3  qts.  1  pt.  2  gills :  how  much  did  they  all 
contain  ? 


1C8  COMPOUND  [SECT.  Vll 

15.  If  a  stage-coach  goes  at  the  rate  of  5  m.  2  fur.  30  r, 
per  hour,  how  far  will  it  go  in  10 hours? 

16.  If  a  Railroad  car  goes  21  m.  2  fur.  10  r.  per  hour 
how  far  will  it  go  in  10  hours  ? 

17.  Bought  12  pieces  of  broadcloth,  each  containing 
27  yds.  1  qr.  2  na. :  how  many  yards  did  all  contain  ? 

18.  If  a  man  mows  3  A.  35  sq.  r.  per  day,  how  many 
acres  can  he  mow  in  30  days  ? 

19.  How  many  square  yards  of  plastering  will  a  house 
which  has  9  rooms  require,  allowing  75  yds.  18  ft.  to  a 
room? 

20.  A  man  bought  15  loads  of  wood,  each  containing 
1  C.  33  ft. :  how  many  cords  did  he  buy  ? 

21.  A  miller  constructed  7  cubical  bins  for  grain,  each 
containing  216  feet  152  in. :  what  was  the  contents  of 
the  whole  ? 

22.  If  a  ship  sails  2°  25'  10"  per  day,  how  far  will  she 
sail  in  20  days  1 

23.  Multiply  56°  42'  11"  by  32. 

24.  If  a  brewer  sells  33  gals.  2  qts.  1  pt.  of  beer  a  day, 
how  much  will  he  sell  in  24  days  ? 

25.  If  a  milk-man  sells  40  gals.  3  qts.    1   pt.  of  milk 
per  day,  how  much  will  he  sell  in  60  days  ? 

26.  What  cost  82  tons  of  iron,  at  £4,  15s.  6-£d.  per- 
ton? 

27.  If  1  acre  produce  33  bu.  2  pks.  5  qts.  of  wheat, 
how  much  will  100  acres  produce  ? 

28.  If  1   suit  of  clothes  requires  9  yds.  3  qrs.  2  na , 
how  much  will  500  suits  require  ? 

29.  If  1  mile  of  Railroad  requires  60  T.  5  cwt.  9  lh> 
of  iron,  how  much  will  50  miles  require  ? 

30.  How  much  wheat  will  it  require  to  make  1000 
barrels  of  flour,  allowing  4  bu.  2  pks.  6  qts.  to  a  barrel  ? 


AHT.  173.]  DIVISION.  169 

DIVISION  OF  COMPOUND  NUMBERS. 

Ex.  1.  Divide  £17,  6s.  9d.  by  4. 

Operation.  Beginning  with  the  pounds 

£       5.       d.    far.          we  find  4  is  contained  in  £17, 
4)17  "  6  "  9  "  0  4  times  and  1  over.     Set  the 

4  "  6  "  8  "   1  4  under  the  pounds,  and  re- 

duce the  remainder  £1  to  shil- 
lings, which  added  to  the  6s.,  make  26s.  4  in  26s. ;  6 
times  and  2s.  over.  Set  the  6  under  the  shillings,  and 
reduce  the  remainder  2s.  to  pence,  which  added  to  the  9d. 
make  -33d.  4  in  33d.,  8  times  and  Id.  over.  Set  the  8 
under  the  pence,  reduce  the  Id.  to  farthings,  and  divide 
as  before.  Ans.  £4,  6s.  8d.  1  far. 

173*  Hence,  we  deduce  the  following  general 
RULE  FOR  DIVIDING  COMPOUND  NUMBERS. 

Begin  with  the  highest  denomination,  and  divide  each 
separately.  Reduce  the  remainder,  if  any,  to  the  next 
lower  denomination,  to  which  add  tJie  number  of  that  de- 
nomination contained  in  the  given  example,  and  divide  the 
sum  as  before.  Proceed  in  this  manner  through  all  the  de- 
nominations. 

OBS.  1.  Each  partial  quotient  will  be  of  the  same  denomination,  as 
that  part  of  the  dividend  from  which  it  arose. 

2.  When  the  divisor  exceeds  12,  and  is  a  composite  number,  it  is 
idvisable  to  divide  first  by  one  factor  and  that  quotient  by  the  other. 
(Art.  78.)     If  the  divisor  exceeds  12,  but  is  not  a  composite  number, 
long  division  may  be  employed.  (Art.  77.) 

3.  The  process  of  dividing  different  denominations,  is  called  Com- 
pound Division. 


QUEST. — 173.  Where  do  you  begin  to  divide  a  compound  number  ? 
What  is  tione  with  the  remainder  ?  Obs.  Of  what  denomination  is  each 
partial  quotient  ?  When  the  divisor  is  a  composite  number,  how  pro- 
ceed ?  What  is  the  process  of  dividing  different  denominations,  called  ? 


170  DECIMAL  [SECT.    VIL 

2.  Divide  £274,  4s.  6d.  by  21. 
Operation. 

£       5.       d. 

3)274  "  4  "  6         Divide  by  the  factors  of  21, 
7)91  "  8"^~2 
13  "  1  "  2  Ans. 

3.  Divide  £635,  17s.  by  31. 

Operation. 

£.      s.     £.  s.              The  remainder  £15,  is  reduced 

31)635,  17  (20.  10-^p    to  shillings,  to  which  we  add  the 

620  given  shillings,  making  31.7,  and 

15  rem.  divide  as  before.     The  remain 

20  der  7s.  may  be  reduced  to  pence 

~o7y  and  divided  again  if  necessary. 

310 
7  rem. 

4.  Divide  £7,  8s.  2d.  by  3. 

5.  Divide  £35,  10s.  8d.  3  far.  by  6. 

6.  Divide  £42,  17s.  3d.  2  far.  by  8. 

7.  A  man  bought  5  cows  for  £23,  16s.  8d. :  how  much 
did  they  cost  apiece  1 

8.  A  merchant  sold  10  rolls  of  carpeting  for  £62,  12s. 
9d. :  how  much  was  that  per  roll  ? 

9.  Paid  £25,   10s.   6^d.  for   12  yards  of  broadcloth: 
what  was  that  per  yard  ? 

10.  A  silver-smith  melted  up  2  Ibs.  8  oz.  10  pwts.  of 
silver,  which   he  made  into   6  spoons:   what  was   the 
weight  of  each  spoon  ? 

11.  The  weight  of  8  silver  tankards  is  10  Ibs.  5  oz. 
7  pwts.  6  grs. :  what  is  the  weight  of  each  ? 

12.  If  8  persons  consume  85  Ibs.  12  oz.  of  meat  in  a 
month,  how  much  is  that  apiece  ? 

13.  A  dairy- wo  man  packed  95  Ibs.  8  oz.  of  butter  in  IG 
boxes :  how  much  did  ea«h  box  contain  ? 

16.  A  tailor  had  76  yds.  2  qrs.  3  na.  of  cloth,  out  of 
which  he  made  8  cloaks  :  how  much  did  each  cloak  con 
tain? 


ARTS.  175,  176.]          FRACTIONS.  171 

17.  A  man  traveled  50  m.  and  32  r.  in  11  hours:  at 
what  rate  did  he  travel  per  hour  ? 

18.  A  man  had  285  bu.  3  pks.  6  qts.  of  grain,  which 
he  wished  to  carry  to  market  in   15  equal  loads :  how 
much  must  he  carry  at  a  load  ? 

19.  A  man  had  80  A.  45  r.  of  land,  which  he  laid  out 
into  36  equal  lots  :  how  much  did  each  lot  contain  ? 


SECTION    VIII. 
DECIMAL    FRACTIONS. 

ART.  175.  When  a  number  or  thing  is  divided  into 
equal  parts,  those  parts  we  have  seen  are  called  Fractions. 
(Art.  105.)  We  have  also  seen  that  these  equal  parts 
take  their  name  or  denomination  from  the  number  of  parts 
into  which  the  integer  or  thing  is  divided.  (Art  103.) 
Thus,  if  a  unit  is  divided  into  10  equal  parts,  the  parts 
are  called  tenths;  if  divided  into  100  equal  parts,  the 
parts  are  called  hundredths;  if  divided  into  1000  equal 
parts,  the  parts  are  called  thousandths,  &c. 

Now  it  is  manifest  that  if  a  tenth  is  divided  into  10 
equal  parts,  1  of  those  parts  will  be  a  hundredth ;  for, 
-rV^-lO^Tuir-  (Art.  138.)  If  a  hundredth  is  divided  into 
10  equal  parts,  1  of -the  parts  will  be  a  thousandth;  for, 
Tifu-^lO^TTMnrj  &c.  Thus  a  new  class  of  fractions  is  ob 
tained,  which  regularly  decreases  in  value  in  a  tenfold 
ratio ;  that  is,  a  class  which  expresses  simply  tenths,  hun- 
dredths, thousandths,  &c.,  without  the  intervening  parts,  as 
m  common  fractions,  and  whose  denominators  are  always 
10,  100,  1000,  &c. 

176.  Fractions  which  decrease  in  a  tenfold  ratio,  or 
>rhich  express  simply  tenths,  hundredths,  thousandths,  &c., 
are  called  DECIMAL  FRACTIONS. 

QUEST.— 175.  What  are  fractions?  From  what  do  the  parts  take 
their  name  ?  176.  What  are  decimal  fractions  *  From  what  do  they 
arise  1  Why  called  decimals  ? 


17'xJ  DECIMAL  [SECT.  VIII 

OBS.  Decimal  fractions  obviously  arise  from  dividing  a  unit  intc 
ten  equal  parts,  then  subdividing  each  of  those  parts  into  ten  other 
equal  parts,  and  so  on.  They  are  called  decimals,  because  they  de- 
crease in  a  tenfold  ratio.  (Art.  10.  Obs.  2.) 

177.  Each  order  of  integers  or  whole  numbers,  it 
has  been  shown,  increases  in  value  from  units  towards  the 
left  in  a  ten-fold  ratio  ;  (Art.  9 ;)  and,  conversely,  each 
order  must  decrease  from  left  to  right  in  the  same  ratio,  till 
we  come  to  units'  place  again. 

178.  By  extending  this  scale  of  notation  below  units 
towards  the  right  hand,  it  is  manifest  that  the  first  place 
on  the  right  of  units,  will  be  ten  times  less  in  value  than 
units'  place  ;  that  the  second  will  be  ten  times  less  than  the 
first ;  the  third  ten  times  less  than  the  second^  &c. 

Thus  we  have  a  series  of  orders  below  units,  which  de- 
crease in  a  tenfold  ratio,  and  exactly  correspond  in  value 
with  tejiths.  hundredths,  thousandths,  &c.,  in  com.  fractions. 

179.  Decimal  Fractions  are  commonly  expressed  by 
writing  the  numerator  with  a  point  (  .  )  before  it. 

OBS.  If  the  numerator  does  not  contain  so  many  figures  as  there 
are  ciphers  in  the  denominator,  the  deficiency  must  be  supplied  by 
prefixing  ciphers  to  it. 

For  example,  -fa  is  written  thus  .  1  ;  -fc  thus  .2  ;  T3u- 
thus  .3  ;  &c.  -j-J-u-  is  written  thus  .01,  putting  the  1  in 
hundredths  place  ;  ifa  thus  .05  ;  &c.  That  is,  tenths  are 
written  in  the  first  place  on  the  right  of  units  ;  hundredths 
in  the  second  place  ;  thousandths  in  the  third  place.  &c. 

180.  The  denominator  of  a  decimal  fraction  is  always 
1  with  as  many  ciphers  annexed  to  it   as  there  are  decimal 
figures  in  the- given  numerator.  (Art.  175.) 

OBS.  The  point  placed  before  decimals,  is  called  the  Decimal  Poinf, 
or  Separatrix.  Its  object  is  to  distinguish  the  fractional  parts  fir  am 
whole  numbers. 


QUEST. — 177.  In  what  manner  do  whole  numbers  increase  and 
decrease  ?  178.  By  extending  this  scale  below  units,  what  -would  be 
the  value  of  the  first  place  on  the  right  of  units  ?  The  second  I  The 
third  ?  With  what  do  these  orders  correspond  ?  179.  How  are  deci- 
mal fractions  expressed.  180.  What  is  the  denominator  of  a  decimal 
fraction  I  Obs.  What  is  the  point  placed  before  decimals  called  ? 


ARTS.  177-183.]  FRACTIONS.  173 

1 8 1  •  The  names  of  the  different  orders  of  decimals 
or  places  below  units,  may  be  easily  learned  from  the 
following 

DECIMAL   TABLE. 


423    .267145986274 

182.  It  will  be  seen  from  this  table  that  the  value 
of  each  figure  in  decimals,  as  well  as  in  whole  numbers, 
depends  upon  the  place  it  occupies,  reckoning  from  units. 
Thus,  if  a  figure  stands  in  the  first  place  on  the  right  of 
units,  it  expresses  tenths ;  if  in  the  second,  kundr&UM,  <fee. 
each  successive  place  or  order  towards  the  right,  decreas- 
ing in  value  in  a  tenfold  ratio.     Hence, 

183.  Each  removal  of  a  decimal  figure  one  place  from 
units  towards  the  right^  diminishes  its  value  ten  times. 

Prefixing  a  cipher,  therefore,  to  a  decimal  diminishes 
its  value  ten  times;  for  H  remove's  the  decimal  one  place 
farther  from  units'  place.  Thus  .4=--fV  ;  but  .04--.;»0 
and  .004=-r,]Vir5  &c. ;  for  the  denominator  to  a  c'ecima.1 
fraction  is  1  with  as  many  ciphers  annexed  to  it,  a&  there 
are  figures  in  the  numerator.  (Art.  180.) 

Annexing  ciphers  to  decimals  does  not  alter  their  value ; 
for,  each  significant  figure  continues  to  occupy  the  same 
place  from  units  as  before.  Thus,  ,5=fa\  so  50=^, 
cr  -fa,  by  dividing  the  numerator;  nd  denominator  by  .10; 
(Art.  11 6;)  and .500=-^%  or  -fa,  fee. 

QUEST. — 181.  Repeat  the  Decimal  Table,  beginning  units,  tenths,  &c. 
182.  Upon  what  does  the  value  of  a  decimal  depend  ?  183.  What  '* 
the  effect  of  removing  a  decimal  figure  one  place  to  the  right?  What 
then  is  tno  effect  of  prefixing  ciphers  to  decimals  ?  What,  of  annexing 
thrall 


174  ADDITION    OF  [SECT.  VIII , 

OBS.  1.  It  should  be  remembered  tha.  the  units'  place  is  alwayg 
the  right  hand  place  of  a  whole  number,  'x  °  effect  of  annexing  and 
prefixing  ciphers  to  decimals,  it  will  be  pere*.  7ed,  is  the  reverse  of 
annexing  and  prefixing  them  to  whole  numbers.  (Art.  58.) 

2.  A  whole  number  and  a  decimal,  written  together,  is  calleu  a 
mixed  number.  (Art.  108.) 

184*  To  read  decimal  fractions. 

Beginning  at  the  left  hand,  read  the  figures  as  if  they 
were  whole  numbers,  and  to  the  last  one  add  the  name  of  it* 
order.  Thus, 

.5  is  read  5  tenths. 

.25  "     «  25  hundredths. 

.324  "     «          324  thousandths. 

.5267  li    "        5267  ten  thousandths. 

.43725  "     "      43725  hundred  thousandths. 

.735168  "     "    735168millionths. 

OBS.  In  reading  decimals  as  well  as  whole  numbers,  the  units' 
place  should  always  be  made  the  starting  point.  It  is  advisable  for 
young  pupils  to  apply  to  every  figure  the  name  of  its  order,  or  the 
place  which  it  occupies,  before  attempting  to  read  them.  Beginning 
at  the  units'  place,  he  should  proceed  towards  the  right,  thus — units, 
tentfis,  hundredUis,  thousandths,  &c.,  pointing  to  each  figure  as  ha 
pronounces  the  name  of  its  order.  In  this  way  he  will  very  soon  bo 
able  to  read  decimals  with  as  much  ease  as  he  can  whole  numbers. 

Read  the  following  numbers : 


(1-) 

.25 

(2.) 
.5317 

(3-) 
3.245 

(4.) 
9.14712 

.362 

.1056 

7.6071 

1.06231 

.451 

.4308 

4.3159 

2.00729 

.5675 

.0105 

3.87816 

9.14051 

.8432 

.0007 

5.91432 

8.06705 

(5.) 
25.02 

(6.) 

56.78417 

(7.) 
1.253456 

(8.) 
2.000008 

36.032 

21.05671 

0.034689 

0.500072 

45.7056 

42.05063 

7.035042 

8.305001 

12.07067 

95.10051 

9.103005 

9.000001 

QUEST. — Obs.  Which  is  the  units'  place  ?  What  is  a  whole  i  umber 
and  a  decimal  written  together,  called  ?  184.  How  are  decimals  read  ? 
06*.  In  reading  decimals,  what  should  be  made  the  starting  point  I 


ARTS,  184-186.]  DECIMALS.  175 

Note.—  Sometimes  we  pronounce  the  word  decimal  when  we  come 
l.o  the  separatrix,  and  then  read  the  figures  as  if  they  were  whole 
numbers;  or,  simply  repeat  them  one  after  another.  Thus,  125.427 
is  read,  one  hundred  twenty-live,  decimal  four  hundred  twenty-seven; 
or,  one  hundred  twenty-five,  decimal  four,  two,  seven. 

Write  the  fractional  parts  of  the  following  numbers  in 
decimals  : 

(9.)  (10.)  (11.) 


•'"lOO  ^  100  1  V  I  00  0 

2  i  5  „  o 


13.  Write  49  hundredths;    3  tenths;   445   ten  thou- 
sandths. 

14.  Write  36  thousandths;  25  hundred  thousandths- 
1  millionth. 

15.  Write  7  hundredths;  3  thousandths;   95  ten  thou- 
sandths ;  63  millionths  ;  26  ten  millionths. 

185.  Decimals  are  Added,  Subtracted,  Multiplied,  and 
Divided,  in  the  same  manner  as  whole  numbers. 

OBS.  The  only  thing  with  which  the  learner  is  likely  to  find  any 
difficulty,  is  pointing  off"  the  answer.  To  this  part  of  the  operation  he 
should  give  particular  attention. 

ADDITION  OP  DECIMAL  FRACTIONS. 

186.  Ex.  1.  What  is  the  sum  of  2.5;  24.457;  123.4 
and  2.369  ? 

Operation,          Write  the  units  under  units,  the  tenths 

2.5  under    tenths,    hundredths    under     hun- 

24.457       dredths,  &c.  ;  then,  beginning  at  the  right 

123.4  hand  or  lowest  order,  proceed   thus:    9 

2.369       thousandths  and  7  thousandths  are  16  thou- 

"152726       san(lths.     Write  the  6  under  the  column 

added,  and  carry  the  1  to  the  next  column 

as  in  addition  of  whole  numbers.   1  to  carry  to  6  hundredthg 

QUEST.  —  Note.    What  other  method  of  reading  decimals  is  men 
lioued  I 


176  ADDITION   OF  [SECT.  VIII. 

makes  7  hundredths  and  5  are  12  hundredths.  Set  the  2 
under  the  column  and  carry  the  1  as  before.  1  to  carry 
to  3  tenths  makes  4,  and  4  are  8  tenths  and  4  are  12 
tenths  and  5  are  17  tenths  or  1  and  7  tenths.  Set  the  7 
under  the  column,  and  carry  the  1  to  the  next  column. 
Finally,  place  the  decimal  point  in  the  amount,  directly  under 
that  in  the  numbers  added, 

187.  Hence,  we  deduce  the  following-  general 
RULE  FOR  ADDITION  OF  DECIMALS. 

Write  the  numbers  so  that  the  same  orders  may  stand  under 
each  other i  placing  tenths  under  tenths,  hundredths  under  hun- 
dredths, <$fc.  Begin  at  the  right  hand  or  lowest  order,  and 
proceed  in  all  respects  as  in  adding  whole  numbers.  (Art.  29.) 

From  the  right  hand  of  the  amount,  point  off  as  many 
figures  for  decimals  as  are  equal  to  the  greatest  number  of  de- 
cimal places  in  either  of  the  given  numbers. 

PROOF. — Addition  of  Decimals  is  proved  in  the  same  mari- 
ner as  Simple  Addition.  (Art.  28.) 

Note. — The  decimal  point  in  the  answer  will  always  fall  directly 
under  the  decimal  points  in  the  given  numbers. 

EXAMPLES. 

(2.)  (3.)  (4.) 

31.25  15.263  20.13 

1.059                      7.0003  117.056 

126.05  213.0507  43.5 

1235.6151                    0.05  2185.05813 

139S9741  An*.  85-306  620.30597 

5.  What  is  the  sum  of  2.5  ;  33.65  and  45.121  ? 

6.  What  is  the  sum  of  65.7  ;  43.09  ;  1.026  and  2.1765 1 


QUEST. — 187.  How  are  decimals  added  1    How  point  off  the  answer ! 
How  is  addition  of  decimals  proved  ? 


ARTS.  187,  188.]  DECIMALS.  177 

7.  What    is  the  sum  of   6.15768;    1.713458   and 
C573128? 

8.  What   is   the  sum  of  .0256 ;  15.6941;  3.856  and 
00035? 

9.  Add  together  256.31  ;  29.7  ;  468.213  ;  5.6  and  .75. 
10.  Add  together  25.61;    78.003;  951.072  and  256 

3052. 
11    Add  together  .567  ;  37.05  ;  63.501  ;  76.25  and  .63. 

12.  Add   together  .005  ;    1.25  ;    6.456 ;  10.2563  and 
15.434. 

13.  Add  together  256.1;   10.15;    27.09;  35.560  and 
2.067. 

14.  Add  together  5.00257;  3.600701  and  2.10607. 

15.  Add  together  5  tenths,  25  hundredths,  566  thou- 
sandths, and  7568  ten  thousandths. 

16.  Add   together  34  hundredths,  67  thousandths,  13 
ten  thousandths,  and  463  millionths. 

17.  Add  together   7  thousandths,  63   hundred   thou- 
sandths, 47  millionths,  and  6  tenths. 

18.  Add 'together  423  ten  millionths,  63  thousandths, 
25  hundredths,  4  tenths,  and  56  ten  thousandths. 

SUBTRACTION  OP   DECIMAL  FRACTIONS. 
188.  Ex.  1.  From  25.367  substract  13.18. 

Operation.  Having   written  the  less  number 

'  under  the  greater,  so  that  units  may 

25.367  stand     under    units,     tenths     under 

tenths,  &c .,  we  proceed  exactly  as  in 
12.187.  Ans.  subtraction  of  whole  numbers.  (Art. 
40.)  Thus,  0  thousandths  from  7 
thousandths  leaves  7  thousandths.  Write  the  7  in  the 
thousandth's  place.  As  the  next  figure  in  the  lower  lino 
is  larger  than  the  one  above  it,  we  borrow  10.  Now  8 
from  16  leaves  8  ;  set  the  8  under  the  column,  and  carry 
1  to  the  next  figure.  (Art.  38.)  Proceed  in  the  same 
manner  with  the  other  figures  in  the  lower  number.  Fi- 
nally, place  the  decimal  point  in  the  remair  Jer  directly 
under  that  in  the  given  numbers. 


178  SUBTRACTION    OF  [SECT.    VEIL 

189.  Hence,  we  deduce  the  following  general 

RULE  FOR  SUBTRACTION  OF  DECIMALS. 

Write  the  less  number  under  the  greater,  with  units  unda 
units,  tenths  wider  tenths,  hundredths  under  hundredths,  <Jr. 
Subtract  as  in  whole  numbers,  and  point  off  the  answer  as  in 
addition  of  decimals.  (Art.  187.) 

PROOF. — Subtraction  of  Decimals  is  proved  in  the  s&mi 
manner  as  Simple  Subtraction.  (Art.  39.) 

Note. — When  there  are  blank  places  on  the  right  hand  of  the  upper 
number,  they  may  be  supplied  by  ciphers  without  altering  the  value 
of  the  decimal.  (Art.  183.) 

EXAMPLES. 

2.  From  15  take  1.5.  Ans.   13.5 

3.  From  256.0315  take  5.641. 

4.  From  15.7  take  1.156. 

5.  From  63.25  take  50. 

6.  From  201.001  take  56.04037. 

7.  From  1  take  .125. 

8.  From  11.1  take  .40005. 

9.  From  .56078  take  .325. 

10.  From  1.66  take  .5589. 

11.  From  3.4001  take  2.000009. 

12.  From  1  take  .000001. 

13.  From  256.31  take  125.4689301. 

14.  From  8960.320507  take  63.001. 

15.  From  57000.000001  take  1000.001. 

16.  From  75  hundredths  take  75  thousandths. 

17.  From  6  thousandths  take  6  millionths. 

18.  From  3252  ten  thousandths  take  3  thousandths 

19.  From  539  take  22  thousandths. 

20.  From  7856  take  236  millionths. 


QUEST. — 189.  How  are  decimals  subtracted  ?    How  point  off  the 
answer  ?    How  is  subtraction  of  decimals  proved  ? 


ARTS.  189-191.]  DECIMALS.  179 

MULTIPLICATION  OF  DECIMAL  FRACTIONS. 
19O.  Ex.  1.  Multiply  .48  by  .5. 

Suggestion. — Multiplying  by  a  fraction,  is  taking  a  part 
of  the  multiplicand  as  many  times  as  there  are  like  parts 
of  o  unit  in  the  multiplier.  (Art.  132.)  Hence,  multiply- 
ing- by  .5,  which  is  equal  to  VV  or  £,  is  taking  half  of  the 
multiplicand  once.  Now  .48,  or  -ffo-i-2=-ftfa.  (Art. 
138.)  But -^=.24.  (Art.  179.) 

Operation.          We  multiply  as  in  whole  numbers,  and 

.48  pointing  off  as  many  decimals  in  the  pro- 

.5  duct  as  there  are  decimal  figures  in  both 

~240  Ans.       factors,  we  have  .240.     But  since  ciphers 

placed  on  the  right  of  decimals  do  not 

affect  t^eir  value,   the  0  may  be  omitted.     (Art.  183.) 

But  ,24=TVo>  which  is  the  same  result  as  before. 

2.  3.  4. 

Multiply    8.45  96.071  456.03 

By                -25  .0032  4.5 

4225  192142  228015 

1690  288213  182412 


Ans.  2.1125  .3074272  2052.135 

191.  From  the  preceding  illustrations  we  deduce 
I'he  following  general 

RULE  FOR  MULTIPLICATION  OF  DECIMALS. 

Multiply  as  in  whole  number 's,  ami  point  off  as  many 
figures  from  the  right  of  the  product  for  decimals,  as  there 
are  decimal  places  both  in  the  multiplier  and  multiplicand. 

If  the  product  does  not  contain  so  many  figures  as  then 
are  decimals  in  both  factors,  supply  the  deficiency  by  prefixing 
ciphers. 


QUEST. — 191.  How  are  decimals  multiplied  together  ?  How  do  you 
point  off  the  product?  When  the  product  does  not  contain  so  many 
figures  as  there  are  decimals  in  both  factors,  what  is  to  be  done  1 


180  MULTIPLICATION   OP  [SECT.  VIIL 

PROOF.  —  Multiplication  of  Decimals  is  proved  in  the  sam 
manner  as  Simple  Multiplication.  (Arts.  53,  74.) 

OBS.  The  reason  for  pointing  off  as  many  decimal  places  in  tae 
product  as  there  are  decimals  in  both  factors,  may  be  illustrated  thus  : 

Suppose  it  is  required  to  multiply  .25  by  .5.  Supplying  the  denom- 
inators .25=1%  and  .5=T5o.  (Art.  180.)  Now 


(Art.  135.)  But  Tffl,=  125;  (Art.  179;)  that  is,  the  product  o! 
.25X-5,  contains  just  as  many  decimals  as  the  factors  themselves.  In 
like  manner  it  may  be  shown  that  the  product  of  any  two  or  more  de- 
cimal numbers,  must  contain  as  many  decimal  figures  as  there  are 
places  of  decimals  in  the  given  factors. 


EXAMPLES. 

Ex.  1.  In  1  piece  of  cloth  there  are  31.7  yards:  how 
many  yards  are  there  in  7.3  pieces  ? 

2.  In  1  barrel  there  are  31.5  gallons:  how  many  gal- 
lons are  there  in  8.25  barrels  ? 

3.  In  one  rod  there  are   16.5  feet:  how  many  feet  are 
there  in  35.75  rods  ? 

4.  How  many  cords  of  wood  are  there  in  45  loads,  al- 
lowing 8.25  of  a  cord  to  a  load  ? 

5.  How  many  rods  are  there  in  a  piece  of  land  25.35 
rods  long,  and  20.5  rods  wide  ? 

6.  If  a  man  can  travel  38.75  miles  per  day,  how  far  can 
he  travel  in  12.25  days? 

7.  How  many  pounds  of  coffee  are  there  in  68  sacks, 
allowing  961.25  pounds  to  a  sack? 

8.  If  a  family  consume  .85  of  a  barrel  of  flour  in  a 
week,  how  much  will  they  consume  in  52.23  week?? 

9-  What  is  the  product  of  10.001  into  .05? 

10.  What  is  the  product  of  50.0065  into  1.003  ? 

192.  When  the  multiplier  is  10, 100, 1000,  &c.,  the 
multiplication  may  be  performed  by  simply  removing  the 
decimal  point  as  many  places  towards  the  right,  as  there 
are  ciphers  in  the  multiplier.  (Arts.  59,  191.) 


QUEST. — How  is  multiplication  of  decimals  proved  ?     192.  How 
proceed  when  the  multiplier  is  10  100,  1000,  &c. 


A.RTS.  192,  193.]  DECIMALS.  181 

11.  Multiply  4.6051  by  100.  Ans   460.51. 

12.  Multiply  2.6501  by  1000. 

13.  Multiply  .5678  by  10000. 

14.  Multiply  .000781  by  2.40001. 

15.  Multiply  1.002003  by  .0024. 

16.  Multiply  .58001  by  .0001003. 

17.  Multiply  8.00 1502  by  .00005. 

18.  Multiply  85689.31  by  .000001. 

19.  Multiply  .0000045  by  69.5. 

20.  Multiply  .0340006  by  .000067. 

21.  Multiply  .5  by  5  millionths. 

22.  Multiply  .15  by  28  ten  thousandths. 

23.  Multiply  25  hundredth  thousandths  by  7.3 

24.  Multiply  225  millionths  by  2.85. 

25.  Multiply  2367  ten  millionths  by  3.0002. 

DIVISION  OP  DECIMAL  FRACTIONS. 
193.  Ex.  1.  Divide  .75  by  .5. 
Operation. 

5). 75  We  divide  as  in  whole  numbers,  and  point 

~L5  Ans.     °ff  1  decimal  figure  in  the  quotient. 

OBS.  We  have  seen  in  the  multiplication  of  decimals,  that  the  pro- 
duct has  as  many  decimal  figures,  as  the  multiplier  and  multiplicand. 
(Art.  191.)  Now  since  the  dividend  is  equal  to  the  product  of  the 
•jivisor  and  quotient,  (Art.  G5,)  it  follows  that  the  dividend  must  have 
as  many  decimals  as  the  divisor  and  quotient  together;  consequently, 
as  the  dividend  has  two  decimals,  and  the  divisor  but  one,  we  must 
point  off  one  in  the  quotient ;  that  is,  we  must  point  off  as  many  de- 
cimals in  the  quotient,  as  the  decimal  places  in  the  dividend  exceed 
those  in  the  divisor. 

2.  Divide  .289  by  2.4. 

Operation. 
2.4).289(.  12+ Ans. 

24  Since  the  divisor  contains  two  figures, 

49"  we  substitute  long  division  for  short, 

48  and  point  off  the  quotient  as  before 

1  rem. 


182  DIVISION  OF  [SECT.  V1IL 

y0te. — When  there  is  a  remainder,  the  sign + should  be  annexed  t« 
«ie  quotient,  to  show  that  it  is  not  complete. 

3.  Divide  1.345  by  .5.  Ans.  2.69. 

4.  Divide  .063  by  9. 

Operation.  In  this  example  the  dividend  has  thre% 

9). 063  more  places  of  decimals  than  the  divisor , 

~007  Ans.    nence  tne  quotient  must  have  three  places 

of  decimals.     We  must,  therefore,  prefix 

two  ciphers  to  the  quotient. 

194.  From  these  illustrations  we  deduce  the  follow- 
ing general 

RULE  FOR  DIVISION  OF  DECIMALS. 

Divide  as  in  whole  numbers,  and  point  off  as  many  fig- 
ures for  decimals  in  the  quotient,  as  the  decimal  places  in  the 
dividend  exceed  those  in  the  divisor.  If  the  quotient  does  not 
contain  figures  enough,  supply  the  deficiency  by  prefixing 
ciphers. 

PROOF. — Division  of  Decimals  is  proved  in  the  same  man- 
ner as  Simple  Division.  (Art.  73.) 

OBS.  1.  When  the  number  of  decimals  in  thedivisor  is  the  same  aa 
that  in  the  dividend,  the  quotient  will  be  a  whole  number. 

2.  When  there  are  more  decimals  in  the  divisor  than  in  the  divi- 
dend, annex  as  many  ciphers  to  the  dividend  as  are  necessary  to  make 
its  decimal  places  equal  to  those  in  the  divisor.     The  quotient  thence 
arising  will  be  a  whole  number.  (Obs.  1.) 

3.  After  all  the  figures  of  the  dividend  are  divided,  if  there  is  a  re- 
mainder, ciphers  may  be  annexed  to  it  and  the  division  continued  at 
pleasure.     The  ciphers  annexed  must  be  regarded  as  decimal  places 
belonging  to  the  dividend. 

Note. — For  ordinary  purposes,  it  will  be  sufficiently  exact  to  carry 
the  quotient  to  three  or  four  places  of  decimals ;  but  when  great  accu- 
racy is  required,  it  must  be  carried  farther. 

QUEST. — 194.  How  are  decimals  divided  ?  How  point  off  the  quo- 
tient ?  How  is  division  of  decimals  proved  ?  Obs.  When  the  number 
of  decimal  places  in  the  divisor  is  equal  to  that  in  the  dividend,  what 
is  the  quotient  ?  When  there  are  more  decimals  in  the  divisor  than  in 
the  dividend,  how  proceed  I  When  there  is  a  remainder,  what  may 
be  done  ? 


A.RTS.  194,  195.]  DECIMALS.  183 


EXAMPLES. 

1.  If  1.7  of  a  yard  of  cloth  will  make  a  coat,  how 
many  coats  will  10.2  yards  make? 

2.  In  6.75  cords  of  wood,  how  many  loads  are  there, 
Allowing  .75  of  a  cord  to  a  load? 

3.  If  a  man  mows   &2  acres  of  grass  per  day,  how 
long  will  it  take  him  to  mow  39.36  acres? 

4.  If  23.25  bushels  of  barley  grow  on  an  acre,  how 
many  acres  will  556  bushels  require  ? 

5.  In  74.25  feet,  how  many  rods  ? 

6.  In  99.225  gallons  of  wine,  how  many  barrels? 

7.  If  a  man  chops  3.75  cords  of  wood  per  day,  how 
many  days  will  it  take  him  to  chop  91.476  cords? 

8.  If  a  man  can  travel  35.4  miles  per  day,  how  long 
will  it  take  him  to  travel  244.26  miles  ? 

9.  A  dairy-man  has  187.5  pounds  of  butter,  which  he 
wishes  to  pack   in  boxes  containing  12.5  pounds  apiece  : 
how  many  boxes  will  it  require  ? 

10.  In  3.575,  how  many  times  .25? 

195.  Whrn  the  divisor  is  10, 100,  1000,  &c.,  the  di 
vision  may  be  performed  by  simply  removing  the  decimal 
point  in  the  dividend  as  many  places  towards  the  left,  as 
there  are  ciphers  in  the  divisor,  and  it  will  be  the  quotient 
required.  (Arts.  80, 194.) 

11.  Divide  756.4  by  100.  Ans.  7.564. 

12.  Divide  1268.2  by  1000.  Ans.  1.2682. 

13.  Divide  1  by  1.25.              14.  Divide  1  by  562.5. 
15.  Divide  .012  by  .005.        16.  Divide  2  by  .0002. 
17.  Divide  5  by  .000001.       18.  Divide  13.2  by  .75. 

19.  Divide  .0248  by  .04. 

20.  Divide  2071.31  by  65.3. 


QUEST.— 195.  When  the  divisor  is  10,  100,  1000,  &c.,  how  may  the 
division  be  performed  ? 


184  REDUCTION   OP  [SECT.  VIIL 

REDUCTION  OF   DECIMALS. 

CASE   I. 
Ex.  1.  Change  the  decimal  .25  to  a  common  fraction. 


Suggestion.  —  Supplying  the  denominator, 
(Art.  180.)  Now  -j2^-  is  expressed  in  the  form  of  a  com 
mon  fraction,  and  as  such  may  be  reduced  to  lower  terms, 
and  be  treated  in  the  same  manner  as  any  other  common 
fraction.  Thus  -?fc=-£u,  or  -J-.  Hence, 

196.  To  reduce  a  Decimal  to  a  Common  Fraction. 

Erase  the  decimal  point  ;  then  write  the  decimal  denomina- 
tor under  the  numerator  ',  and  it  will  form  a  common  fraction^ 
which  may  be  treated  in  tJie  same  manner  as  other  common 
fractions. 

2.  Change  .125  to  a  common  fraction,  and  reduce  it  to 
'.he  lowest  terms.  Ans.  -J-. 

3.  Reduce  .66  to  a  common  fraction,  &c. 

4.  Reduce  .75  to  a  common  fraction,  &c. 

5.  Reduce  .375  to  a  common  fraction,  &c. 

6.  Reduce  .525  to  a  common  fraction,  &c. 

7.  Reduce  .025  to  a  common  fraction,  &c. 

8.  Reduce  .875  to  a  common  fraction,  &c. 

9.  Reduce  .0625  to  a  common  fraction,  &c. 
10.  Reduce  .000005  to  a  common  fraction,  &c. 

CASE    II. 
Ex.  1.  Change  %  to  a  decimal. 

Suggestion.  —  Multiplying  both  terms  by  10  the  fraction 
becomes  f-ft.  As^ain  dividing  both  terms  by  5,  it  becomes 
-tV  (Art.  116.)  "But  ^  -.6,  (Art.  179,)  which  is  the 
decimal  required. 


QUEST.—196.  How  are  Decimals  reduced  to  Common  Fractions  ? 


ARTS.  196,  197.]  DECIMALS.  185 

Now  since  we  make  no  use  of  the  denominator  10 
after  it  is  obtained,  we  may  omit  the  process  of  getting  it ; 
for  if  we  annex  a  cipher  to  the  numerator  and  divide  it 
by  5,  we  shall  obtain  the  same  result. 

Operation. 

5)3.0  A  decimal  point  is  prefixed  to  the  quo- 

.6         tient,  to  distinguish  it  from  a  whole  number. 

PROOF. — .6  reduced  to  a  common  fraction  is  ^ ;  (Art 
196;)  andTV-f.  (Art.  120.) 
2.  Reduce  4-  to  a  decimal. 


8)1.000  Annex   ciphers  to  the  numerator  and 

125        proceed  as  before.     Hence, 

197.  To  reduce  a  Common  Fraction  to  a  Decimal. 

Annex  ciphers  to  the  numerator  and  divide  it  by  the  de- 
nominator. Point  off  as  many  decimal  figures  in  the  quo- 
tient, as  you  have  annexed  ciphers  to  the  numerator. 

OBS.  1.  If  there  are  not  as  many  figures  in  the  quotient  as  you 
have  annexed  ciphers  to  the  numerator,  supply  the  deficiency  by  pre- 
fixing ciphers  to  the  quotient. 

2.  The  reason  of  this  process  may  be  illustrated  thus.  Annexing  a 
cipher  to  the  numerator  multiplies  the  fraction  by  10.  (Arts.  59, 133.) 
If,  therefore,  the  numerator  with  a  cipher  annexed  to  it,  is  divided  by 
the  denominator,  the  quotient  will  obviously  be  ten  times  too  large. 
Hence,  in  order  to  obtain  the  true  quotient,  or  a  decimal  equal  to  the 
given  fraction,  the  quotient  thus  obtained  must  be  divided  by  10,  which 
is  done  by  pointing  off  one  figure.  (Art.  80.)  Annexing  2  ciphers  to 
the  numerator  multiplies  the  fraction  by  100;  annexing  3  cipners  by 
1000,  &c.,  consequently,  when  2  ciphers  are  annexed,  the  quotient 
will  be  100  times  too  large,  and  must  therefore  be  divided  by  100; 
when  three  ciphers  are  annexed,  the  quotient  will  be  1000  times  too 
large,  and  must  be  divided  by  1000;  &c.  (Art.  80.) 


QUEST. — 197.  How  are  Common  Fractions  reduced  to  Decimals? 
Obs.  When  there  are  not  so  many  figures  in  the  quotient  as  you  have 
fcnnexed  ciphers,  what  is  to  be  done  I 


186  REDUCTION  07         [SECT.  VIII 

3.  Reduce  f  to  decimals.  Ans.  1.5. 

4.  Reduce  •£-,  and  %  to  decimals. 

5.  Reduce  -^-,  and  -fe  to  decimals. 

6.  Reduce  -f.  -|,  and  f  to  decimals. 

7.  Reduce  -f-,  £,  and  -^  to  decimals. 

8.  Reduce  -^V,  ^  and  ^  to  decimals. 

9.  Reduce  -f,  -f,  and  -fa  to  decimals. 

10.  Reduce  4^,  and  ^  758  5  to  decimals. 

1 1.  Reduce  -g^-.  and  -nfW  to  decimals. 

12.  Reduce  i  to  a  decimal.  .Arcs.  .333333-f-. 

13.  Reduce  iff  to  a  decimal.        Ans.  .128128128+. 

198.  It  will  be  seen  that  the  last  two  examples  can 
not  be  exactly  reduced  to  decimals ;  for  there  will  continue 
to  be  a  remainder  after  each  division,  as  long  as  we  con- 
tinue the  operation. 

In  the  12th,  the  remainder  is  always  1 ;  in  the  13th, 
after  obtaining  three  figures  in  the  quotient,  the  remainder 
is  the  same  as  the  given  numerator,  and  the  next  three 
figures  in  the  quotient  are  the  same  as  the  first  three, 
when  the  same  remainder  will  recur  again. 

The  same  remainders,  and  consequently  the  same  fig- 
ures in  the  quotient,  will  thus  continue  to  recur,  as  long 
as  the  operation  is  continued. 

199.  Decimals  which  consist  of  the  same  figure  or 
set  of  figures  continually  repeated,  as  in  the  last  two  ex- 
amples, are  called  Periodical  or   Circulating   Decimals; 
also,  Repeating  Decimals^  or  Repetends. 

CASE    III. 
Ex.  1.  Reduce  7s.  6d.  to  the  decimal  of  a  pound. 

Suggestion. — First,  reduce  7s.  6d.  to  pence  for  the  nu- 
merator, and  £1  to  pence  for  the  denominator  of  a  com 

QUEST.— 199.  What  are  Periodical  01  Repeating  Decimals  ? 


ARTS.  198-200.]  DECIMALS.  137 

mon  fraction,  and  we  have  ££fo.  (Art.  165.)     Now  -fft 
reduced  to  a  decimal  is  £.375.  Ans.     Hence, 

2OO.  To  reduce  a  compound  number  to  the  decimal 
of  a  higher  denomination. 

First  reduce  the  given  compound  number  to  a  common  frac- 
tion ;  (Art.  165  ;)  then  reduce  tJie  common  fraction  to  a  de- 
cimal. (Art.  197.) 

2.  Reduce  5s.  4d.  to  the  decimal  of  £1. 

Ans.  £.2666+. 

3.  Reduce  15s.  6d.  to  the  decimal  of  £1. 

4.  Reduce  12s.  6d.  to  the  decimal  of  £1. 

5.  Reduce  9d.  to  the  decimal  of  1  shilling. 

6.  Reduce  7d.  2  far.  to  the  decimal  of  a  shilling. 

7.  Reduce  1  pt.  to  the  decimal  of  a  quart. 

8.  Reduce  18  hours  to  the  decimal  of  a  day. 

9.  Reduce  9  in.  to  the  decimal  of  a  yard. 

10.  Reduce  2  ft.  6  in.  to  the  decimal  of  a  yard. 

11.  Reduce  6  furlongs  to  the  decimal  of  a  mile. 

12.  Reduce  13  oz.  8  dr.  to  the  decimal  of  a  pound. 

CASE   IV. 
Ex.  1.  Reduce  £.123  to  shillings,  pence,  and  farthings. 

Operation.          Multiply  the  given  decimal  by  20,  as 

P  -,9o        if  it  were  a  whole  pound,  because  20s. 

~20      make  £1>  and  point  off  as  many  figures 

for  decimals,  as  there  are  decimal  places 

shil.  2.460      in    the   multiplier    and    multiplicand. 

12      (Art.  191.)     The  product  is  in  shillings 

pence  5.520      arjd   a  decimal  of   a   shilling.     Then 

4      multiply  the  decimal  of  a  shilling  by 

far.  2.080      12,  and  point  off  as  before,  &c.     The 

numbers   on  the  left  of  the   decimals, 

Ans.  2s.  5d.  2  f.    viz :  2s.  5d.  2  far.  form  the  answer. 

Hence, 


QUEST. — 200.  How  is  a  compound  number  reduced  to  the  decimal  of 
a  higner  denomination  ? 


188  FEDERAL  [SECT.    VIIL 

20 1.  To   reduce  a  decimal   compound   number  to 
whole  numbers  of  lower  denominations. 

Multiply  the  given  decimal  by  that  number  which  it  takes 
of  the  next  lower  denomination  to  make  ONE  of  this  higher,  as 
in  reduction,  (Art.  161, 1,)  and  point  off  the  product,  as  in 
multiplication  of  decimal  fractions.  (Art.  191.)  Proceed  in 
this  manner  with  the  decimal  figures  of  each  succeeding  pro- 
duct,  and  the  numbers  on  the  left  of  the  decimal  point  in  th( 
several  products,  will  constitute  the  whole,  number  required. 

2.  Reduce  £.125  to  shillings  and  pence.     Ans.  2s.  6d. 

3.  Reduce  .625s.  to  pence  and  farthings. 

4.  Reduce  £.4625  to  shillings  and  pence. 

5.  Reduce  .756  gallons  to  quarts  and  pints. 

6.  Reduce  .6254  days  to  hours,  minutes,  and  seconds, 

7.  Reduce  .856  cwt.  to  quarters,  &c. 

8.  Reduce  .6945  of  a  ton  to  hundreds,  &c. 

9.  Reduce  .7582  of  a  bushel  to  pecks,  &c. 

10.  Reduce  .8237  of  a  mile  to  furlongs,  &c. 

11.  Reduce  .45683  of  an  acre  to  roods  and  rods. 

12.  Reduce  .75631  of  a  yard  to  quarters  and  nails, 

FEDERAL  MONEY. 

202.  FEDERAL  MONEY  is  the  currency  of  the  United 
States.     The  denominations  are,  Eagles,  Dollars,  Dimes] 
Cents,  and  Mills. 

TABLE. 

10  mills  (m.)  make  1  cent,  marked  ct. 

10  cents  "  1  dime,      "       d. 

10  dimes  "  1  dollar,     "       doll,  or  $. 

10  dollars  "  1  eagle,      "      E. 

OBS.  Federal  Money  was  established  by  Congress,  Aug.  8th,  1786. 
Previous  to  this,  English  or  sterling  money  was  the  principal  curren^ 
cy  of  the  country. 


QUEST. — 201.  How  are  decimal  compound  numbers  reduced  to  whole 
ones?  202.  What  is  Federal  Money  ?  Recite  the  Table.  Obs.  When 
and  by  whom  was  it  established  ? 


ARTS.  201-204.]  MONEY.  189 

Note.— Many  foreign  coins  are  still  in  circulation.  Indeed  some  of 
the  rates  of  postage  established  by  the  government,  were,  until  re- 
cently, adapted  to  foreign  coins.  To  the  28th  Congress  belongs  the 
honor  of  abolishing  these  anti-national  rates,  and  of  establishing  others 
in  Federal  Money. 

203.  The  national  coins  of. the  United  States  are  of 
three  kinds,  viz  :  gold,  silver,  and  copper. 

1.  The  gold  coins  are  the  eagle,  the  double  eagle  *  half 
tagle,  quarter  eagle,  and  gold  dollar  * 

The  eagle  contains  258  grains  of  standard  gold  ;  the  dou- 
ole  eagle,  half  eagle,  and  quarter  eagle,  like  proportions. 

2.  The  silver  coins  are  the  dollar,  half  dollar,  quarter 
dollar,  the  dime,  and  half  dime. 

The  dollar  contains  412£  grains  of  standard  silver;  the 
others,  like  proportions. 

3.  The  copper  coins  are  the  cent,  and  half  cent. 

The  cent  contains  168  grains  of  pure  copper  ;  the  half 
cent,  a  like  proportion. 
Mills  are  not  coined. 

OBS.  1.  The  fineness  of  gold  used  for  coin,  jewelry,  and  other  pur- 
poses, also  the  gold  of  commerce,  is  estimated  by  the  number  of  parts 
of  gold  which  it  contains.  Pure  gold  is  commonly  supposed  to  be 
divided  into  24  equal  parts,  called  carats.  Hence,  if  it  contains  10 
parts  of  alloy,  or  some  baser  metal,  it  is  said  to  be  14  carats  fine ;  if  5 
parts  of  alloy,  19  carats  fine ;  and  when  absolutely  pure,  it  is  24  car- 
ats fine.-}- 

2.  The  present  standard  for  both  gold  and  silver  coins  of  the  United 
States,  by  Act  of  Congress,  1837,  is  900  parts  of  pure  metal  by  weight 
to  100  parts  of  alloy.  The  alloy  of  gold  coin  is  composed  of  silver 
and  copper,  the  silver  not  to  exceed  the  copper  in  weight.  The  alloy 
of  silver  coin  is  pure  copper. 

20 4,  All  accounts  in  the  United  States  are  kept  in 

QUEST. — 203.  Of  how  many  kinds  are  the  coins  of  the  United  States  1 
What  are  they  ?  What  are  the  gold  coins  ?  The  silver  coins  ?  The 
topper  ?  Obs.  How  is  the  fineness  of  gold  estimated  ?  Into  how  many 
carats  is  pure  gold  supposed  to  be  divided  ?  When  it  contains  10  parts 
of  alloy,  how  fine  is  it  said  to  be  I  5  parts  of  alloy  ?  2  parts  ?  4  parts  ? 
What  is  the  standard  for  the  gold  and  silver  coins  of  the  United  States  * 
What  is  the  alloy  of  gold  coins  ?  What  of  silver  coins  ?  204.  In  what 
»re  accounts  kept  ?  How  would  you  express  5  eagles  ?  7  E.  and  5 
lolls.  ?  10  E.  ?  How  express  6  dimes  ?  8  dimes  ?  10  dimes  ? 
*  By  Act  of  Congress,  Feb.  20th.  1849.  *  SillimHn's  Chemistry. 


190  FEDERAL  [SECT.  VIIL 

dollars,  cents,  and  mills.  Eagles  are  expressed  in  dollars, 
and  dimes  in  cents.  Thus,  instead  of  5  eagles,  we  say, 
50  dollars ;  instead  of  7  eagles  and  5  dollars,  we  say,  75 
dollars,  &c.  So,  instead  of  6  dimes,  we  say,  60  cents ; 
instead  of  8  dimes  and  7  cents,  we  say,  87  cents,  &c. 

2O 5.  It  will  be  seen  from  the  Table  that  Federal 
Money  is  based  upon  the  Decimal  system  of  Notation ; 
that  its  denominations  increase  and  decrease  from  right  to 
left  and  left  to  right  in  a  tenfold  ratio,  like  whole  num- 
bers and  decimals. 

2OG.  The  Dollar  is  regarded  as  the  unit;  cents  and 
mills  are  fractional  parts  of  the  dollar,  and  are  distin- 
guished from  it  by  a  decimal  point  or  separatrix  (.)  in  the 
same  manner  as  common  decimals.  (Art.  179.)  Dollars 
therefore  occupy  units'  place  of  simple  numbers ;  eagles, 
or  tens  of  dollars,  tens'1  place,  &c.  Dimes,  or  tenths  of  L 
dollar,  occupy  the  place  of  tenths  in  decimals ;  cents  or 
hundredths  of  a  dollar,  the  place  of  hundredths ;  mills,  or 
thousandths  of  a  dollar,  the  place  of  thousandths ;  tenths 
of  a  mill,  or  ten  thousandths  'of  a  dollar,  the  place  of  ten 
thousandths,  &c. 

OBS.  1.  Since  dimes  in  business  transactions  are  expressed  in  cents, 
two  places  of  decimals  are  assigned  to  cents.  If  therefore  the  number 
of  cents  is  less  than  10,  a  cipher  must  always  be  placed  on  the  left  hand 
of  them;  for  cents  are  hundredths  of  a  dollar,  and  hundredths  occupy 
the  second  decimal  place.  (Art.  181.)  For  example,  4  cents  are 
written  thus  .04;  7  cents  thus  .07;  9  cents  thus  .09,  &c. 

2.  Mills  occupy  the  third  place  of  decimals;  for  they  are  thou- 
sandths of  a  dollar.  Consequently,  when  there  are  no  cents  in  the 
given  sum,  two  ciphers  must  be  placed  before  the  mills.  Hence, 

2O 7 .  To  read  any  sum  of  Federal  Money. 

Call  all  the  figures  on  the  left  of  the  decimal  point  fol 
lars  ;  the  first  two  figures  after  the  point,  are  cents ,  tkt 

QUEST. — 205,  How  do  the.  denominations  of  Federal  Money  increase 
and  decrease  7  Upon  what  is  it  based  ?  206.  What  ;s  regarded  as  the 
unit  in  Federal  Money  ?  What  are  cents  and  mills  ?  How  are  they 
distinguished  from  dollars  ?  207.  How  do  you  read  Federal  Money  ? 
Obs.  What  other  mode  of  reading  Federal  Money  i 


ARTS.  205-207.]  MONEY.  191 

third  figure  denotes  mills  ;  the  other  places  on  the.  right  are 
decimals  of  a  mill.  Thus,  $3.25232  is  read,  3  dollars,  25 
cents,  2  mills,  and  32  hundredths  of  a  mill. 

OBS.  Sometimes  all  the  figures  after  the  point  are  read  as  decimals 
of  a  dollar.  Thus,  $5.356  is  read,  "  5  and  356  thousandths  dollars." 

Read  the  following  sums  of  Federal  Money : 

1.  2.  3. 

$250.56  $44.081  $3.7542 

105.863                 60.05  0.6054 

200.057                 75.003  4.0151 

506.507                 20.501  6.0057 

850.071                 30.065  8.0106 

Write  the  following,  sums  in  Federal  Money : 

4.  63  dollars,  and  85  cents.  Ans.  $63.85. 

5.  150  dollars,  and  73  cents. 

6.  201  dollars,  and  9  cents. 

7.  300  dollars,  5  cents,  and  3  mills. 

8.  4  dollars,  6  cents,  and  8  mills. 

9.  100  dollars,  7  cents,  5  mills,  and  3  tenths  of  a  mill. 
10.  1000  dollars,  6  mills,  and  36  hundredths  of  a  mill. 

Note. — In  business  transactions,  when  dollars  and  cents  are  ex- 
pressed together,  the  cents  are  frequently  written  in  the  form  of  a 
common  fraction.  Thus,  $"76.45  are  written  76-A-5.-  dollars. 

REDUCTION  OF  FEDERAL  MONEY. 

-CASE    I. 
Ex.  1.  How  many  cents  are  there  in  75  dollars  ? 

Suggestion. — L.'nce  in  1  dollar  there  are  100  cents,  in 
75  dollars  there  are  75  times  as  many.  And  75x100- 
7500.  Ans.  7500  cents. 

2.  In  9  cents,  how  many  mills  1          Ans.  90  mills. 

3.  In  25  dollars,  how  many  mills  ?  Ans,  25000  mills. 


192  FEDERAL  [SECT.  VIII 

Note. — To  multiply  by  10,  100,  &c.,  is  simply  annexing  as  many 
ciphers  to  the  multiplicand,  as  there  are  ciphers  in  the  multiplier. 
(Art.  59.)  Hence, 

208.  To  reduce  dollars  to  cents,  annex  two  ciphers. 
To  reduce  dollars  to  mitts,  annex  three  ciphers. 

To  reduce  cents  to  mills,  annex  one  cipher. 

OBS.  To  reduce  dollars  and  cents  to  cents,  erase  the  sign  of  dottart 
and  the  separatrix.  Thus,  $25.36  reduced  to  cents,  becomes  2536 
cents. 

4.  In  $5,  how  many  cents  ? 

5.  How  many  mills  in  $364  ? 

6.  How  many  mills  in  $621  ? 

7.  How  many  cents  in  $6245  ? 

8.  Reduce  $75.26  to  cents. 
Q    Reduce  $625.48  to  cents. 

CASE    1 1 ." 

10.  In  4500  cents,  how  many  dollars? 

Suggestion. — Since  100  cents  make  1  dollar,  4500  cents 
will  make  as  many  dollars  as  100  is  contained  times  in 
4500.  And  4500+ 100=45.  Ans.  $45. 

11.  In  150  mills,  how  many  cents'?     Ans.  15  cents. 

12.  In  25000  mills,  how  many  dollars  ?     Ans.  $25. 

Note. — To  divide  by  10,  100,  &c.,  is  simply  cutting  off  as  man/ 
figures  from  the  right  of  the  dividend  as  there  are  ciphers  in  the  Di- 
visor. (Art.  80.)  Hence, 

209.  To  reduce  cents  to  dollars,  cut  off  two  figures  on 
the  right. 

To  reduce  mills  to  dollars,  cut  off  three  figures  on  tht 
right. 

To  reduce  mills  to  cents,  cut  off  one  figure  on  the  right. 

OBS.  The  figures  cut  off  are  cents  and  mills. 


QUEST. — 208.  How  are  dollars  reduced  to  cents  ?  Dollars  to  mills  ? 
Cents  to  mills  ?  Obs.  Dollars  and  cents  to  cents  ?  209.  How  are  cents 
reduced  to  dollars  ?  Mills  to  dollars  ?  Mills  to  cents  ?  Obs.  What 
are  the  figures  cut.  off? 


ARTS.  208-2  i  1  .]  MONEY.  193 

13.  In  325  cents,  how  many  dollars  ?       Ans.  $3.25. 

14.  In  423  mills,  how  many  cents  ?     Ans.  42c.  3m. 

15.  In  4320  mills,  how  many  dollars  ? 
16    How  many  dollars  in  63500  cents  ? 
17.  How  many  cents  in  4890  mills  ? 

2  1  0.  Since  Federal  Money  is  expressed  according  to 
the  decimal  system  of  notation,  it  is  evident  that  it  may  be 
subjected  to  the  same  operations  and  treated  in  the  same 
manner  as  decimal  fractions. 

ADDITION  OF  FEDERAL  MONEY. 

Ex.  1.  A  man  bought  a  cow  for  $15.75,  a  calf  for 
$2.375,  a  sheep  for  $3.875,  and  a  load  of  hay  for  $8.68  • 
how  much  did  he  pay  for  all  ? 


Operation  ^e  wr*te  tne  dollars  under  dol- 

lars, cents  under  cents,  &c.     Then 

$15.75^  a^    each    column    separately,   and 

2.375  point  off  as  many  figures  for  cents 

3-87^,  and  mills,  in  the  amount,  as  there  are 

places  of  cents  and  mills  in  either  of 

$30.680  Ans.        the  given  numbers. 

211*  Hence,  we  derive  the  following  general 

RULE  FOR  ADDING  FEDERAL  MONEY. 

Write,  dollars  under  dollars,  cents  under  cents,  fyc.,  50 
that  the  same  orders  or  denominations  may  stand  under  each 
other.  Add  each  column  separately,  and  point  of  the  amount 
as  in  addition  of  decimal  fracti'MS.  (Art  187.) 

OBS.  If  either  of  the  given  numbers  have  no  cents  expressed,  it  i« 
customary  to  supply  their  place  by  ciphers. 

2.  A  farmer  sold  a  firkin  of  butter  for  $9.28,  a  cheese 
for  $1.17,  a  quarter  of  veal  for  56  cents,  and  a  bushel  of 
wheat  for  $1.  12  :  how  much  did  he  receive  for  the  whole  ? 


QUEST.— 211.  How  is  Federal  Money  added?    How  point  off  the 
amount  ?    Obs.  When  any  of  the  given  numbers  hare  no  cents  ex- 
1,  how  is  their  place  supplied  ? 

7 


i94  FEDERAL  [SECT.  VIII 

3.  A  man  bought  a  hat  for  $5.375,  a  cloak  for  $35.68, 
and  a  pair  of  boots  for  $4.75  :  how  much  did  he  pay  fof 
all? 

4.  What  is  the  sum  of  $37.565,  $85:20,  $90.03,  and 
$150.638? 

5.  What  is  the  sum  of  $10.385,  $46.238,   $190.62 
and  $23.036? 

6.  What   is  the  sum  of  $23.005,  $16.03,  $110.738, 
and  $131.26? 

7.  What  is  the  sum  of  63  dolls,  and  4  cts.,  86  dolls,  and 
10  cts.,  and  47  dolls,  and  37  cts.  ? 

8.  What  is  the  sum  of  $608.05,  $365.205,  $2.268,  and 
$47.006? 

9.  What  is  the  amount  of  1 1  dolls.   3  cts.  and  5  mills, 
16  dolls,  and  8  mills,  49  dolls.  7  cts.  and  8  mills? 

10.  What  is  the  amount  of  100  dolls,  and  61   cts.,  51 
dolls,  and  3  cts.,  65  dolls.  8  cts.  and  3  mills  ? 

1 1.  What  is  the  amount  of  95  dolls.  67  cts.  and  8  mills, 
1 20  dolls.  45  cts.,  101  dolls.  7  cts.  and  9  mills? 

12.  A  lady  bought  a  bonnet  for  $6.67,  a  pair  of  gloves 
for  $0.625,  a  pair  of  shell  combs  for  $0.75,  and  a  cap  for 
$2.50 :  what  was  the  amount  of  her  bill  ? 

SUBTRACTION  OP  FEDERAL  MONEY. 

Ex.  1.  A  man  bought  a  horse  for  $56.50,  and  a  cow 
for  $23.38  :  how  much  more  did  he  pay  for  his  horse  than 
his  cow  ? 

Operation  ^e  wr^te  tne  ^ess  number  under 

'  the    greater,   placing    dollars    under 

dollars,  &c.,  then  subtract,  and  point 

off  the  answer   as   in   subtraction  of 

$33.12  Ans.     decimals. 

• 
212*  Henee,  we  derive  the  following  general 

RULE  FOR  SUBTRACTING  FEDERAL  MONEY. 

Write  the  less  number  under  the  greater,  with  dollars  undci 
dollars,  cents  under  cents,  fyc.,  then  subtract,  and  point  ojf 
the  remainder  as  in  subtraction  of  decimal  fractions.  (Art 
189.) 


ARTS.  2 1 2,  2 1 3.]  MONET.  195 

OBS.  If  either  of  the  given  numbers  have  no  cents  expressed,  it  b 
customary  to  supply  their  place  by  ciphers. 

2.  A  man  owing  $57.35,  paid  $17.93 :  how  much  does 
he  still  owe  ?  Ans.  $39.42.      . 

3.  A   grocer  bought  two  hogsheads   of  molasses  for 
$68.90,  and  sold  it  for  $79.26 :  how  much  did  he  gain 
by  the  bargain  ? 

4.  A  man  owed  a  debt  of  $105,  and  paid  all  but  $23. 
67  :  how  many  dollars  did  he  pay  ? 

5.  A  merchant  bought  a  quantity  of  silks  for  $237.63. 
and  sold  it  for  $196.03  :  how  much  did  he  lose? 

6.  A  drover  bought  a  flock  of  sheep  for  $357,  and  sold 
them  for  $17.33  less  than  he  paid  for  them:  how  much 
did  he  sell  them  for  ? 

7.  What  is  the  difference  between  365  dolls.  7  cts.  and 
208  dolls.  20  cts.  ? 

8.  From  1  cent  subtract  6  mills. 

9.  From  1  dollar,  6  cts.  and  7  mills,  take  89  cts.  and  3 
mills. 

10.  From  96  dollars,  6  cents,  take  41  dolls.,  63  cents, 
and  8  mills. 

11.  From  100  dollars,  10  cents,  and  3  mills,  take  1  cent 
and  5  mills. 

12.  From  1000  dollars,  6  cents,  take  100  dolls,  and  5 
mills. 


MULTIPLICATION  OF  FEDERAL  MONET. 

213,  In  Multiplication  of  Federal  Money,  as  well  as 
in  simple  numbers,  the  multiplier  must  always  be  consid- 
ered an  abstract  number.  (Art.  45.  Obs.  2.) 

Ex  1.  How  much  will  5  yards  of  cloth  cost,  at  $1.75 
per  yard  ? 


QUEST.— 212.  How  is  Federal  Money  8ubtracted  ?  How  point  off 
the  remainder  t  Obs.  When  either  of  the  given  numbers  have  no  cent* 
expressed,  how  is  their  place  supplied  ?  213.  Jn  Multiplication  of  Ffe. 
d«ral  Money,  what  must  one  of  the  given  factors  be  considered  ? 


196  FEDERAL  [SECT.  VIIL 

Operation         If  1  yard  cost  $1.75,  5  yards  will  obvious. 

$1.75         ly  cost  5  times  as  much.     Hence,  we  multi 

5          ply  the  price   of   1   yard  by  the  number  ol 

$8.75  Ans. .7ar^s?  an^  point  off  two  figures  for  decimals 

'in  the  product.  (Art.  191.) 

2.  How  much  will  15.8  yards  of  fringe  cost,  at  12  cents 
per  yard  ? 

Operation.  Reasoning  as  before,  15.8  yards  will  cost  15.8 
15.8  times  12  cents.  But  in  performing  the  mul- 
.1 2  tiplication,  it  is  more  convenient  to  take 

$1  89l5     ^e    ^   f°r   tne    multiplier5  and  tf16  result 

will  be  the  same  as  if  it  was  placed  for  the 

multiplicand.  (Art.  47.)     Point  off  the  product  as  before. 

214.  Hence,  when  the  price  of  one  article,  one  pound, 
one  yard,  &c.,  is  given  to  find  the  cost  of  any  number  ol 
articles,  pounds,  yards,  &c. 

Multiply  the  price  of  one  article  and  the  number  of  articles 
together,  and  point  off  the  product  as  in  multiplication  of 
decimals.  (Art.  191.) 

3.  Multiply  $45.035  by  6.2.  Ans.  $279.217. 

215*  From  the  preceding  illustrations  we  derive  the 
following  general 

RULE  FOR  MULTIPLYING  FEDERAL  MONEY. 

Multiply  as  in  simple  numbers,  and  point  off  the  product 
as  in  multiplication  of  decimal  fractions.  (Art.  191.) 

OBS.  1.  When  the  price  or  the  quantity  contains  a  common  frac- 
tion, the  fraction  should  be  changed  to  a  common  decimal.  (Art.  197.) 

2.  In  business  operations,  when  the  mills  in  the  answer  are  5,  01 
over,  it  is  customary  to  call  them  a  cent;  when  under  5,  they  ar« 
disregarded. 

QUEST.— 214.  When  the  price  of  1  article,  1  pound,  &c.,  is  given, 
how  is  the  cost  of  any  number  of  articles  found  ?  215.  What  is  the  rul« 
for  Multiplication  of  Federal  Money  ?  Obs.  When  the  price  or  quan- 
«ty  contains  a  common  fraction,  what  should  b§  don«  with  it! 


ARTS.  214,  215.]  MONEY.  197 

4.  What  will  10  Ibs.  of  beef  cost,  at  6£  cents  a  pound? 

Solution.—  6%  cts.=.065,  and  .065x1  0=.65. 

Ans.  65  cents. 

5.  What  cost  14  Ibs.  of  starch,  at  10-J-  cts.  per  pound? 

6.  WThat  cost  15^  pounds  of  sugar,  at  9£  cts.  a  pound? 

7.  What  cost  25  gals,  of  molasses,  at  18-f-  cts.  a  gallon? 

8.  What  cost  23-J-  Ibs.  of  raisins,  at  8£  cts.  per  pound  ? 

9.  What  cost  33£  Ibs.  of  candles,  at  12^  cts.  per  pound? 

10.  What  cost  16f  Ibs.  of  hyson  tea,  at  56^  cts.  a  pound? 

11.  What  will  83  Ibs.  of  beef  cost,  at  $4.62i  per  hund.  ? 

Analysis.  —  83  pounds  are  i3^-  of  100  pounds;  there 
fore  83   pounds  will  cost  -ffo  of  $4.625;  and  ffa  of 


Operation.  We  multiply  the  price  of  100 

$4.625  ($4.625)  by  83,  the  given  num- 

83  ber  of  pounds,  and  the  product 

-  13875  $383.875,  is  the  cost  of  83  Ibs.  at 

3  70  00  $4.625  ^QI  pound.    But  the  price 

—  ---  is  $4.625   per  hundred;   conse 

$3.83  875  Ans.       quently,  the  product  $383.875  is 

100  tin:es  too  large,  and  must 

therefore  be  divided  by  100,  to  give  the  true  answer. 
But  to  divide  by  100,  we  simply  remove  the  decimal 
point  two  places  toward  the  left.  (Art.  195.) 

12.  What  will  825  feet  of  boards  cost,  at  $6.75  per 
1000? 

Reasoning  as  before,  825  feet  will 
cost  ja^  of  $6.75.  We  multiply  the 
price  of  1000  feet  by  the  given  number 
of  feet,  and  divide  the  product  by  1000. 
To  divide  by  1000,  we  remove  the  de- 
cimal point  three  places  towards  the 

56875  left"  (Art  195'>     Hence> 


198  FEDERAL  [SECT.  VIIL 

216.  To  find  the  cost  of  articles  bought  and  sold  by 
the  100,  or  1000. 

Multiply  the  given  price  by  the  given  number  of  articles , 
then  if  the  price  is  for  100,  divide  the  product  by  100  ;  but  ij 
ike  price  is  for  1000,  divide  it  by  1000.  (Art.  195.) 

13.  At  $4.50  per  1000,  what  will  1250  bricks  cost? 

14.  A  farmer  sold  a  quarter  of  beef,  weighing-  256.5 
pounds,  at  $5.37£  per  100  :  how  much  did  he  receive  for 
it? 

15.  At  $4.62£  per  hundred,  what  will  1675  pounds  of 
pork  cost  ? 

16.  What  cost  2129  feet  of  spruce  boards,  at  $18.25 
per  1000? 

17.  How  much  will  456f  yards  of  shirting  cost,  at  12-J- 
cts.  per  yard  ? 

18.  What  cost  156  Ibs.  of  chocolate,  at   15£  cents  a 
pound  ? 

19.  What  cost  235  pounds  of  cheese,  at  6-J-  cents  a 
pound  ? 

20.  What  cost  175  dozens  of  eggs,  at  10£  cents  per 
dozen? 

21.  At  47£  cents  per  bushel,  what  will  be  the  cost  oi 
300  bushels  of  corn  ? 

22.  What  will   153  Ibs.  of  sugar  cost,  at  8£  cents  per 
pound  ? 

23.  What  will    1500  pounds  of  butter  cost,  at  $8.50 
per  hundred? 

24.  What   cost  28500   feet  of  timber,   at   $3.76  per 
100? 

25.  What  cost  8230  feet  of  mahogany,  at  $70.20  per 
1000? 

26.  What  cost  7630  hemlock  shingles,  at  $3.50  per 
1 000  ? 

27.  What  cost  15024  pine  shingles,  at  $8.37  per  1000  ? 

28.  At   16i  cts.  a  pound,  what  will  219^  pounds  of 
honey  cost  ? 

QUEST. — 216.  How  do  you  find  the  cost  of  articles  bought  and  »W 
by  the  100,  or  1000? 


ARTS,  216,  217.]  MONEY.  199 

29.  At  $2.67-f  per  yard,  what  will  400  yards  of  cloth 
eost? 

30.  At  $5f  per  barrel,  what  will  1560  barrels  of  flour 

EOSt? 

DIVISION  OF  FEDERAL  MONEY. 

Ex.  1.  A  man  bought  6  hats  for  $25.68:  how  much 
did  they  cost  apiece  ? 

Operation.  If  6  hats  cost  $25.68,  1  hat  will  cost 

6)25.68  one  sixth  of  $25.68.     Divide  as  in  sim- 

$4.28  Ans.      P^e  numbers,  and  point  off  two  decimal 
figures  in  the  quotient.  (Art.  194.) 

Proof. 

$4.28  If  1  hat  costs  $4.28,  6  hats  will  cost  6  times 

6  as  much ;  and  $4.28x6=$25.68,  which  is  the 

$25^68  £iven  cost-     Hence, 

217*  When  the  number  of  articles,  pounds,  yards, 
&c.,  and  the  cost  of  the  whole  are  given,  to  find  the  price 
of  one  article,  one  pound,  &c. 

Divide  the  whole  cost  by  the  whole  number  of  articles,  and 
point  off"  the  quotient  as  in  division  of  decimal  fractions. 
(Art.  194.) 

2.  How  marly  yards  of  cloth,  at  $3.13  per  yard,  can 
be  bought  for  $20.345  ? 

Operation.  Since   $3.13   will   buy    1    yard 

3  loxonq/is/A  s  A        $20.345  will  buy  as  many  yards 

}?878  (  as    ®3'13    is    contained    times    in 

$20.345.      Divide    as    in    simple 

numbers,  and  point  off  one  decimal 
figure  in  the  quotient.  (Art.  194.) 
Proo/— $3.13x6.5=$20.345.     Hence, 

QUEST.— 217.  When  the  number  of  articles,  pounds,  &o.,  and  the 
«o»t  of  the  whole  are  given,  how  is  the  cost  of  one  article  found  ? 


200  FEDERAL  [SECT. 

218.  When  the  price  of  one  article,  pound,  yard, 
find  the  cost  of  the  whole  are  given,  to  find  the  number  oi 
articles,  &c. 

Divide  the  whole  cost  by  the  price  of  one,  and  point  offtht 
quotient  as  in  Art.  217. 

3.  Divide  $149.625  by  $2.375.  Ans.  63. 

4.  If  $75  are  divided  equally  among  18  men,  how 
much  will  each  receive  ? 

Operation. 

18)75($4.166  Ans.         After  dividing  the  $75   by   18, 
72  there  is  a  remainder  of  3  dollars, 

3QQQ  which  must  be  reduced  to  cents  and 

18  mills,  (Art.  208,)  and  then  be  di- 

vided as  before.     The  ciphers  thus 
annexed  must  be  regarded  as  deci- 

mals;  consequently   there  will  be 

120  three  decimal  figures  in  the  quo- 

108  tient. 


12  rem. 

219*  From  the  preceding  illustrations  we  derive  the 
following  general 

RULE  FOR  DIVIDING  FEDERAL  MONEY. 

Divide  as  in  simple  numbers,  and  point  oj[jhe  quotient  as 
in  division  of  decimal  fractions.  (Art.  194jp 

OBS.  After  all  the  figures  of  the  dividend  are  divided,  if  there  is  a 
remainder,  ciphers  may  be  annexed  to  it,  and  the  operation  may  be 
continued  as  in  division  of  decimals.  (Art.  194.  Obs.  3.)  The  ciphers 
thus  annexed  must  be  regarded  as  decimal  places  of  the  dividend. 

5.  How  many  pounds  of  cheese,  at  7  cts.  a  pound,  can 
you  buy  for  $1.47  ? 

QUEST.— 218.  When  the  price  of  1  article,  1  pound,  &c.,  and  the 
cost  of  the  whole  are  given,  how  is  the  number  of  articles  found  ? 
219.  What  is  the  rule  for  Division  of  Federal  Money  ?  Obs.  When 
there  i»  a  remainder  after  all  the  figures  of  the  dividend  are  divided, 
how  proceed '? 


ARTS.  218,  219.]  MONEY.  201 

6.  A  man  paid  $0.75  for  the  use  of  a  horse  and  buggy 
to  go  8  miles :  how  much  was  that  per  mile  ? 

7.  How  many  quarts  of  cherries,  at  7  cents  a  quart,  can 
you  buy  for  $1.12? 

8.  How  many  pounds  of  figs,  at  14  cents  a  pound,  can 
you  buy  for  $3.57  ? 

9.  How  many  watermelons,  at  12-J-  cts.  apiece,  can  be 
bought  for  $3  ? 

10.  How  many  pen-knives,  at  20  cts.  apiece,  can  be 
oought  for  $7.20? 

11.  At  1 7-£  cts.  a  quart,  how  many  quarts  of  molasses 
can  be  bought  for  $4.40? 

12.  A  man  bought  50  pair  of  thick  boots  for  $175' 
how  much  did  he  give  a  pair  ? 

13.  A  man  paid  $485.50  for  260  sheep:  how  much 
did  he  give  per  head  ? 

14.  At  $2.50  a  cord,  how  many  cords  of  wood  can  T 
buy  for  $165? 

15.  At  $4.75  per  barrel,  how  many  barrels  of  flour 
can  I  buy  for  $8.50  ? 

16.  If  a  man's  income  is  $1.68  per  day,  how  much  is 
it  per  hour  ? 

17.  If  a  man  pays  $3. 62-£  per  week  for  board,  how 
long  can  he  board  for  $188.50  ? 

1 8.  Suppose  a  man's  income  is  $500  a  year,  how  much 
is  that  per  day  ? 

19.  Suppose  a  man's  interest  money  is  $28.80  per  day 
how  much  is  it  per  minute  ? 

20.  A  mason  received  $94.375  for  doing  a  job,  which 
took  him  75£  days  :  how  much  did  he  receive  per  day  ? 

21.  At  $1.1 2^-  per  bushel,  how  many  bushels  of  wheat 
can  be  bought  for  $523.75  ? 

22.  If   $1285.20   were   divided   equally  among    125 
men,  what  would  each  receive  ? 

23.  If  $1637.10  were  divided  equally  among  150  men, 
what  would  each  receive  ? 

24.  The  salary  of  the  President  of  the  United  States 
js  $25000  a  year :  how  much  does  he  receive  per  day? 


202  BILLSL  [SECT.  VIII 

APPLICATIONS   OF  FEDERAL  MONEY. 
BILLS,   ACCOUNTS,    &C. 

2 2O.  A  Bill,  in  mercantile  operations,  is  a  paper 
containing  a  written  statement  of  the  items,  and  the  pric* 
or  amount  of  goods  sold. 

Ex.  1.  What  is  the  cost  of  the  several  articles,  and 
what  the  amount,  of  the  following  bill  ? 

BOSTON,  May  25th,  1845. 
James  Brown,  Esq. 

Bought  of  Fair/kid  $  Lincoln, 

5  yds.  Broadcloth,  at         $3.25 
3  yds.  Cambric,  "  .12£ 

3  doz.  Buttons,  "  .15 

6  skeins  Sewing  Silk,       "  .06| 

4  yds.  Wadding,  «  .08 

Amount,  $17.77. 
Received  Pay't, 

Fairfield  tSf  Lincoln. 


(2.) 

NEW  HAVEN,  Sept.  2d,  1845. 
Hon.  R.  S.  Baldwin. 

Bought  of  Durrie  fy  Peckt 

4  Lo veil's  Young  Speaker,  at  $  .62£ 

5  Olmsted's  Rudiments,  "  .58 

6  Morse's  Geography.  "  .50 
8  Webster's  Spelling  Book,  "  .10 
3  Day's  Algebra,  "  1.25 

What  was  the  cost  of  the  several  articles,  and  what  tin 
amourt  of  his  bill  ? 


ART.  220.]  BILLS.  203 

(3.) 

NEW  YORK,  Aug.  18th,  1845. 
John  Jacob  Aslor,  Esq. 

Bought  of  G.  W  Lewis  <£  Co 


25  Ibs.  Sugar, 
50  Ibs.  Coffee, 

at 
« 

$.09     - 
.11     - 

12  Ibs.  Tea, 

u 

.75     - 

14  Ibs.  Raisins, 

u 

.14     - 

9  doz.  Eggs, 

a 

.10     - 

15  Ibs.  Butter, 

u 

.12*- 

What  was  the  cost  of  the  several  articles,  and  what 
the  amount  of  his  bill  ?  f 

(4.) 

PHILADELPHIA,  June  3d,  1845. 
W.  A.  Sanford,  Esq. 

To  James  Conrad,  Dr. 
For  28  yds.  Silk,  at  SI.  25 

«    22  yds.  Muslin,  «       .56 

"    16  pair  Cotton  Hose,         «        .37-i-      - 
«    35    "     Silk          «  "1.10 

«    25    «     Shoes,  «      1.25 

Wha   was  the  cost  of  the  several  articles,  and  haw 
much  is  due  on  his  account  1 


(5.) 

CINCINNATI,  July  1st,  1845. 
Messrs.  Holmes  fy  Homer 

.    To  H.  W.  Morgan  4*  Co.,  Dr. 
For  100  bbls.  Flour,        at  $4.50 
«      50     «      Pork,          «     8.25 
"      25     "      Be"ef,  «     9.75 

«    112  kegs  Xard,  «     3.25 

«      25  bush. 'Corn,          «       .34' 

What  was  the  cost  of  the  several  articles,  and  how 
ir.uch  is  due  on  his  account  ? 


204  PERCENTAGE.  [SECT.  IX, 

(6.) 

NEW  ORLEANS,  Aug.  12th,  1845. 
F.  C.  Emerson,  Esq. 

To  W.  H.  Arnold  $  Co.,  Dr. 
For       35  hhds.  Molasses,    at   $12.60 
«     2100  Ibs.  Sugar,  «          .05£      - 

"  14000  Ibs.  Cotton,  «          .07*      • 

"     1350  Ibs.  Coffee,  "          .06}      - 

«  31200  Ibs.  Rice,  «          .08 

rt       150  boxes  Oranges,      «        4.12£      - 

CREDIT. 

By  500  Clocks,  at      $5.00 

"   Note  to  balance  account, 

What  was  the  amount  of  charges,  and  what  the  amoun 
of  the  note  ? 


SECTION    IX. 

PERCENTAGE. 

ART.  222.  The  terms  Percentage  and  Per  Cent,  signi- 
fy a  certain  allowance  on  a  hundred  ;  that  is,  a  certain  parl 
of  a  hundred,  or  simply  hundredths.  Thus  the  expres- 
sions 2  per  cent.,  4  per  cent,  6  per  cent.,  &c..  of  any 
number  or  sum  of  money,  signify  2  hundredths  (TOT:) 
4  hundredths  (TOT?)  6  hundredths  (TOT?)  &c-  °f  that  num- 
ber or  sum.  For  example, 

1  per  cent,  of  $100,  is  ^fa  of  that  sum,  which  is  1  dollar; 

2  per  cent,  of  $100,  is  -T-2_  of  that  sum,  which  is  2  dollars; 
4  per  cent,  of  $100,  is  -p^-  of  that  sum,  which  is  4  dollars; 

6  per  cent,  of  $100,  is  -^fo  of  that  sum,  which  is  6  dollars,  &c. 
Hence,  universally, 

QUEST — 222.  What  do  the  terms  percentage  and  per  cent,  signify  I 
What  is  meant  by  2  per  cent. ,  4  per  cent. ,  &c. ,  of  any  sum  ?  What  then 
does  any  given  percentage  of  any  number  or  sum  of  money  imply  1  Obs, 
From  what  are  the  terms  percentage  and  per  cent,  derived  * 


ARTS.  222,  222. a.]      PERCENTAGE.  205 

222.  ft.  Any  given  percentage  of  any  number,  or  sum  of 
money,  implies  so  many  units  for  every  100  units;  so  many 
dollars  for  every  100  dollars;  so  many  cents  for  every  100 
cents;  so  many  pounds  for  every  100  pounds,  <Sfc. 

Note. — The  terms  Percentage  and  Per  Cent,  are  derived  from  the 
Latin  per  and  centum,  signifying  by  the  hundred. 

MENTAL    EXERCISES. 

Ex.  1.  A  boy  found  a  purse  containing  8  dollars,  and 
on  returning  it,  the  owner  gave  him  6  per  cent,  of  the 
money :  how  much  did  the  boy  receive  ? 

Suggestion. — 6  per  cent,  is  6  cents  for  every  100  cents. 
(Art.  222.)  If,  then,  he  received  6  cents  for  1  dollar, 
(100  cents,)  for  8  dollars,  he  must  have  received  8  times  6 
cents,  or  48  cents.  Ans.  48  cents. 

2.  What  is  6  per  cent,  of  10  dollars  ?     Ans.  60  cents. 

3.  The  printer's  boy  collected  12  dollars  on  some  bills, 
for  which  his  employer  gave  him  4  per  cent,  for  his  ser- 
vices :  how  much  did  he  receive  1 

4.  What  is  4  per  cent,  of  8  dollars?    6  dollars?    10 
dollars?  7  dollars?   12  dollars?   15  dollars? 

5.  A  man  borrowed  12  dollars  for  a  year,  and  agreed  to 
give  6  per  cent,  for  the  use  of  it :  how  much  did  he  pay  ? 

6.  What  is  6  per  cent,   of  10  dollars?    4  dollars?    6 
dollars?  8  dollars?   11  dollars?   15  dollars? 

7.  A  market-man  sold  20  dollars'  worth  of  butter  foi 
one  of  his  neighbors,  who  paid  him   5   per  cent,  for  his 
trouble  :  how  much  did  he  receive  ? 

8.  What  is  5  percent,  of  12  dollars?    10  dollars?  7 
dollars?   15  dollars?  20  dollars? 

9.  A  farmer  bought  a  cow  for  12  dollars,  and  sold  it 
again  so  that  he  gained  10  per  cent,  by  his  bargain :  how 
much  did  he  gaki  ? 

10.  What  is  10  per  cent,  of  12  dollars?  20  dollars?   14 
dollars?   18  dollars?   13  dollars?  15  dollars? 

11.  What  is   11   per  cent,  of  1   dollar?  4  dollars?   10 
dollars?  7  dollars?  6  dollars?  9  dollars?   12  dollars? 


206  PERCENTAGE,  [SECT.  IX. 

12.  A  boy  having  100  canary  birds,  lost  7  per  cent,  of 
fliem  by  disease  :  how  many  cf  them  did  he  lose  ? 

13.  What  is  7  per  cent,  of  200  dollars?  300  dollars? 
500?  900?  700?  , 

Suggestion. — Since  7  per  cent,  is  7  dollars  on  100  dol- 
lars, for  200  dollars  it  is  twice  as  much,  or  14  dollars,  &c. 

14.  A  merchant  invested  500  dollars  in  a  certain  ad- 
venture, and  gained  6  per  cent,  on  the  sum  invested  : 
now  much  did  he  gain  ? 

15.  What  is  3  per  cent,  of  100  dollars?  300?  500? 
600?  900? 

16.  What  is  5  percent,  of  200  dollars?  400?  300? 
700?  1000?  800?  1100? 

17.  What  is  8  per  cent,  of  200  dollars?  500?  600? 
300?  700?  400?  1200? 

18.  What  is  6  per  cent,  of  300  dollars?  500?  400? 
700?   1100?  900?  1200? 

19.  What  is  7  per  cent,  of  400  dollars?  300?  500? 
900?  600?  1000? 

20.  What  is   1 0  per  cent,  of  500  dollars  ?  300?  700? 
1200?  1500?  2000? 

21.  What  is  8  per  cent,  of  200  dollars?  400?  800? 
900?  1000? 

22.  What  is  9  per  cent,  of  300  dollars?  600?  500? 
800?  700? 

23.  What  is   12  per  cent,  of  8  dollars?  7  dollars?  6 
dollars  ? 

24.  What  is  20  per  cent,  of  4  dollars?  5  dollars?  6 
dollars  ? 

25.  What  is  15  per  cent,  of  3  dollars?  4  dollars?  6 
dollars? 

26.  What  is   18  per  cent,  of  3  dollars?  5  dollars?  4 
dollars  ? 

27.  What  is  25  percent  of  4  dollars?  5  dollars?  & 
dollars  ? 

28.  What  is  30  per  cent,  of  3  dollars?  5  dollars?  9 
iollars? 


.  223.] 


PERCENTAG] 


SOT 


jp  EXERCISES   FOR    THE   SLATE. 

223.  We  have  seen  that  hundredths  are  decimal  ex 
pressions,  occupying  the  first  two  places  of  figures  on  the 
right  of  the  decimal  point.  (Arts.  181,  182.)  Now,  since 
percentage  and  per  cent,  signify  hundredths,  it  is  manifest 
they  may  properly  be  expressed  by  decimals. 

PERCENTAGE    TABLE. 


1  per  cent. 

is  written  thu*  : 

.01 

2  per  cent 

. 

« 

U                <( 

.02 

3  per  cent. 

. 

u 

U             « 

.03 

4  per  cent. 

. 

" 

•  1               M 

.04 

6  per  cent. 

«         « 

u 

(i               t< 

.06 

7  per  cent. 

.        . 

M 

U               « 

.07 

10  per  cent. 

.        .        . 

U 

u          a 

.10 

12  per  cent. 

. 

(( 

u          « 

.12 

25  per  cent. 

.        . 

1 

(i          « 

.25 

50  per  cent. 

. 

« 

<4              i« 

.50 

75  per  cent. 

• 

n 

M               (< 

.75 

99  per  cent. 

. 

• 

«          a 

.99 

100  per  cent. 

. 

n 

M               M 

1.00 

103  per  cent. 

.        . 

u 

<£                (i 

1.03 

125  per  cent.,  &c. 

.        . 

(i 

«(                (t 

1.25 

i  per  cent.,  that  is, 
4  per  cent.,  that  is, 
|  per  cent.,  that  is, 

£  of  1  per  cent, 
i  of  I  per  cent, 
i  of  1  per  cent. 

1C 

(( 
(( 

It                « 
(T               « 
It              tt 

.005 
.0025 
.0075 

13£  per  cent. 

. 

N 

«               (t 

.13125 

25  1  per  cent. 

It 

tt               (1 

.25375 

OBS.  1.  It  will  be  seen  from  the  preceding  Table,  that  when  the 
given  per  cent,  is  less  than  10,  a  cipher  must  be  prefixed  to  the  figure 
expressing  it,  in  the  same  manner  as  when  the  number  of  cents  ia 
less  than  10.  (Art.  206.  Obs.  1.) 

When  the  given  per  cent,  is  more  than  100,  it  must  plainly  require 
a  mixed  number  to  express  it.  (Art.  183.  Obs.  2.) 

2.  Parts  of  1  per  cent,  may  be  expressed  either  by  a  common  frac- 
tion, or  by  decimals.     Thus,  the  expression  174  per  cent.,  is  equiva- 
lent  to  .17625  per  cent. 

3.  The  Jirst  two  decimal  figures  properly  denote  the  per  cent.,  for 
they  are  hundredths  ;  the  other  decimals  being  parts  of  hundredth»t 
express  parts  of  1  per  cent. 


QUEST. — 223.  How  may  percentage  or  per  cent,  be  expressed  !  06*. 
When  the  given  per  cent.  IB  less  than  10,  how  i»  ii  written  I  Wh*a 
taore  than  1UO,  how  t 


208  PERCENTAGE.  [SECT.  IX 

1.  Write  1  per  cent.,  2  per  cent.,  3  per  cent.,  5  pel 
cent.,  8  per  cent.,  and  9  per  cent.,  in  decimals. 

2.  Write  13  per  cent.  ;  15;  30;  50;  75;  49;  73;  85 

3.  Write  i  per  cent.  ;  *;  *;  £  ;  f;  -ft;  ij  ij  il  * 

4.  Write  3-^  per  cent.;  5$;  16*;  125;  33  H;  462£. 

5.  A  merchant  handed  some  bills  amounting  to  $400 
to  a  constable,  and  gave  him  3  per  cent,  for  collecting 
them  :  how  much  did  the  constable  receive  for  his  ser- 
vices? 


Analysis.  —  Since  3  per  cent,  is  T^,  the  constable  must 
have  received  -fa  of  $400,  or  3  dollars  for  every  100  dol- 
lars he  collected.  Now  -rfo  of  $400  is  f££,  which  is  equal 
to  $4  ;  and  3  hunclredths  is  3  times  $4,  or  $12. 

Operation.  Since  -ri^—  .03,  we  have  only  to  mul- 

$400  tiply  the  given  number  of  dollars  by  .03 

.03  and  it  will  give  the  answer  in  cents, 

$12~00.  Ans      whicn  must  be  reduced  to  dollars  by 

pointing  off  2  decimals.     (Art.  209.) 

Note.  —  It  is  important  for  the  learner  to  observe,  that  the  amount  of 
money  collected,  is  made  the  basis  upon  which  the  percentage  is  com- 
puted. That  is,  the  constable  is  entitled  to  3  dollars,  as  often  as  he 
collects  100  dollars,  and  not  as  often  as  he  pays  over  100  dollars,  as  is 
frequently  supposed.  For  in  the  latter  case  he  would  receive  only 
TuT,  instead  of  -j-^j-  of  the  sum  in  question.  This  distinction  is  im- 
portant, especially  in  calculating  percentage  on  large  sums.  Hence, 

225.  To  calculate  percentage  on  any  number  or  sura 
of  money. 

Multiply  the  given  number  or  sum  by  the  given  per  cent 
expressed  decimally;  and  point  off  the  product  as  in  multi- 
plication of  decimal  fractions.  (  Art.  191.) 

OBS.  If  the  per  cent,  contains  a  common  fraction  which  cannot  b« 
expressed  decimally,  first  multiply  by  the  decimal,  then  by  the  com- 
mon fraction  of  the  given  per  cent.,  and  point  off  the  sum  of  their  pro- 
ducts as  above. 

QUEST.  —  Note.  When  it  is  said  that  a  man  receives  a  certain  per 
cent,  for  collecting  money,  upon  what  is  the  per  cent,  calculated  ? 
225.  How  do  you  calculate  percentage  ?  Obs.  If  the  per  cent-  contains 
a  common  fraction  which  cannot  be  expressed  decimally,  how  proceed  1 


ART.  225.]  PERCENTAGE.  209 

6.  What  is  2  per  cent,  of  $350?  Ans.  $7. 

7.  What  is  4  per  cent,  of  $63  ?  Ans.  $2.52. 

8.  What  is  3  per  cent,  of  $145.25?     Ans.  $4.3575. 

9.  What  is  i  per  cent,  of  $180.42  ?  (Art.  223.  Obs.  2.) 
10.  What  is  i  per  cent,  of  $827.63  ? 

1  1.  What  is  f  per  cent,  of  $128.632? 

12.  What  is  £  per  cent.f  o  $90.45  ? 

13.  What  is  10  per  cent,  of  $600.451  ? 

14.  What  is  12  per  cent,  of  $2500.63  ? 

15.  What  is  20  per  cent,  of  $2250.84? 

16.  What  is  3£  per  cent,  of  $436  ? 

Suggestion.  —  3|per  ct.  is  equal  to  .032.  (Art.  223.  Obs.  2.) 
17.  What  is  2|  per  cent,  of  $144? 

Operation. 

$144  Since  %  per  cent,  cannot  be  ex- 

.02-^-  actly  expressed   by   decimals,   we 

$2.88=2  per  ct.  first  multiply  by  .02,  and  then  by  -J-  : 

48—^-  per  cent.  (Art.  134.  a  ;)  and  point  off  two 

<ft7r77g  j  decimal  places.  (Art.  225.  Obs.) 


18.  What  is  4£  per  cent,  of  $257  ? 

19.  What  is  8-f-  per  cent,  of  $673  ? 

20.  A  merchant  having  deposited  $200  in  a  bank,  af- 
terwards drew  out  10-£  per  cent,  of  it:  how  many  dollars 
did  he  draw  out? 

21.  A  merchant  makes  a  deposit  of  $1864  and  draws 
out  25  per  cent,  of  it  :  how  much  has  he  left  in  the  bank  ? 

22.  A  merchant  shipped  865  boxes  of  lemons  ;  on  the 
passage  home,  15  per  cent,  of  them  were  thrown  over- 
board :  how  many  boxes  did  he  lose  ;  and  how  many  had 
he  left? 

23.  How  much  is  6£  per  cent,  of  $1000? 

24.  How  much  is  7  per  cent,  of  $1526  33  ? 

25.  How  much  is  8£  per  cent  of  $16.325? 

26.  A  young  man  worth  $1,500,  lost  31-J  per  cent  of 
it  in  gambling  :  how  much  did  ho  lose  ;  and  how  much 
had  he  left? 


210  PERCENTAGE.  [SECT.  I3L 

27.  A  merchant  bought  a  cargo  of  llour  for  $1230,  and 
paid  41  per  cent,  for  bringing  it  home :  what  was  the 
whole  cost  of  his  flour  ? 

28.  What  is  371  per  cent,  of  $100  ?     Of  $2537.50  ? 

29.  What  is  1 12  per  cent,  of  $  150  ? 

30.  What  is  125  per  cent  of  $635  ? 

31.  What  is  250  per  cent,  of  $17.35? 

32.  Which  is  the  most,  7  per  cent,  of  $1000,  or  6  per 
cent,  of  $1100? 

33.  What  is  the  difference  between  6  per  cent,  and  7 
percent,  of  $12000? 

34.  What  is   the   difference  between  9  per  cent,  of 
$2000,  and  6  per  cent,  of  $3000  ? 

35.  What  is  17£  per  cent,  of  $10000? 

36.  What  is  201  per  cent,  of  $10500? 

37.  A  man  gave  his  two  sons  $10000  apiece ;  the  elder 
added  151  per  cent,  to  his  the  first  year,  ancHhe  younger 
spent  15^-  Per  cent-   of  his:  what  was  the  difference  of 
their  property  at  the  end  of  the  first  year  ? 

38.  A  labouring  man  earning  $225  a  year,  laid  up  231 
per  cent,  of  it :  how  much  did  he  spend  ? 

39.  A  man  having  deposited  $856.25  in  a  savings  bank, 
drew  out  31-^  per  cent,  of  it:  how  much  had  he  left  in 
the  bank  ? 

40.  A  farmer  owning  3560  sheep,  lost  50  per  cent,  of 
them  by  disease :  how  many  had  he  left  ? 

APPLICATIONS  OF  PERCENTAGE. 

226.  PERCENTAGE,  or  the  method  of  reckoning  by 
hundredths,  is  applied  to  various  calculations  in  the  practical 
concerns  of  life.  Among  the  most  important  of  these  are 
Commission,  Brokerage,  the  Rise  and  Fall  of  Stocks, 
Interest,  Discount,  Insurance,  Profit  and  Loss,  Duties, 
and  Taxes.  Its  principles,  therefore,  should  be  thorough- 
ly understood  by  every  scholar. 


QUEST. — 226.  To  what  are  the  principles  of  percentage   applied! 
What  are  some  of  the  most  important  of  these  calculations  I 


ARTS.  226-231.]          COMMISSION,  211 

COMMISSION,  BROKERAGE,  AND  STOCKS. 

227.  Commission  is  the  per  cent,  or  sum  charged  by 
fcgents  for  their  services  in  buying  and  selling  goods,  or 
transacting  other  business. 

OBS.  An  Agent  who  buys  and  sells  goods  for  another,  is  called  a 
Commission  Merchant,  a  Factor,  or  Correspondent. 

228*  Brokerage  is  the  per  cent,  or  sum  charged  by 
money  dealers,  called  Brokers,  for  negotiating  Bills  oj 
Exchange,  and  other  monetary  operations,  and  is  of  the 
same  nature  as  Commission. 

229*  By  the  term  Stocks,  is  meant  the  Capital  01 
moneyed  institutions,  as  incorporated  Banks,  Manufacto- 
ries, Railroad  and  Insurance  Companies  ;  also,  the  funds 
of  Government,  State  Bonds,  &c. 

OBS.  Stocks  are  usually  divided  into  portions  of  $100  each,  called 
thares;  and  the  owners  of  these  shares  are  called  Stockholders. 

230.  The  original  cost  or  valuation  of  a  share  is 
called  its  nominal,  or  par  value ;  the  sum  for  which  it  can 
be  sold,  is  its  real  value,  which  varies  at  different  times. 

OBS.  1.  Th*e  rise  or  fall  of  Stocks  is  reckoned  at  a  certain  per 
cent,  of  its  par  value.  The  term  par  is  a  Latin  word,  which  signifies 
equal,  or  a  state  of  equality. 

2.  When  stocks  sell  for  their  original  cost  or  valuation,  they  ars 
said  to  be  at  par ;  when  they  sell  for  more  than  cost,  they  are  said  to 
be  above  par,  or  at  an  advance ;  when  they  do  not  sell  at  cost,  they  are 
said  to  be  below  par,  or  at  a  discount. 

3.  Persons  who  deal  in  Stocks  are  usually  called  Stock  Brokers,  or 
Stock  Jobbers. 

231.  The  commission  or  allowance  made  to  factors 
and  brokers,  and  the  rise  and  fall  of  stocks,  are  usually 
reckoned  at  a  certain  percentage  on  the  amount  of  money 

QUEST. — 227.  What  is  commission  ?  Obs.  What  is  an  agent  who 
ouys  and  sells  goods  for  anothej  usually  called  ?  228.  What  is  bro- 
kerage? 229.  What  is  meant  by  stocks?  How  are  stocks  usually 
divided  ?  Waat  are  the  owners  of  the  shares  called  ?  230.  What  is 
the  par  value  of  stocks  ?  What  the  real  value  ?  Obs.  What  is  the 
meaning  of  the  term  par  ?  When  are  stocks  at  par  ?  When  above 
par  ?  When  below  ?  231.  How  are  commission,  brokerage,  and  the 
me  or  tail  of  stocks  reckoned  ? 


212  BROKERAGE.  [SECT.  IX 

employed  in  the  transaction,  or  on  the  par  value  of  th« 
given  shares.     Hence, 

232.  To  compute  commission,  brokerage,  and  th« 
advance  or  discount  on  stocks. 

Multiply  the  given  sum  by  the  given  per  cent,  expressed  in 
decimals,  and  point  off  the  product  as  in  Percentage. 

When  the  commission  is  to  be  deducted  in  advance  from  a 
specified  sum  and  the  balance  invested,  divide  the  given  amouni 
by  $1  increased  by  the  per  cent,  commission,  and  the  quotient 
will  be  the  part  to  be  invested.  Subtract  the  part  invested 
from  the  given  amount,  and  the  remainder  will  be  the  com- 
mission required. 

EXAMPLES. 

Ex.  1.  A  commission  merchant  sold  a  quantity  of  coro 
for  $236,  and  charged  2  percent,  commission:  how  much 
did  he  receive  for  his  services  ?  Ans.  $4.72. 

2.  A  tax-gatherer  collected  $533.56,  for  which  he  was 
to  have  3  per  cent,  commission  :  what  did  he  receive  ? 

3.  An  auctioneer  sold  $860.45  worth  of  goods,  at  2} 
per  cent,  commission  :  how  much  did  he  receive  ? 

4.  An  agent  sold  a  quantity  of  oil  for  $265.35,  and 
charged  2^-  per  cent,  commission:    how  much  did  the 
agent  receive ;  and  how  much  the  .owner  ? 

5.  Sold  goods  to  the  amount  of  $356,  at  4-$-  per  cent, 
commission  :  how  much  did  I  receive  for  my  services  ? 

6.  Bought  goods  amounting  to  $480,  at  3-^  per  cent, 
commission  :  what  is  the  amount  of  my  commission  ? 

7.  What  is  the  commission  on  $163.625,  at  6-J-  per  ct.  ? 

8.  What  is  the  commission,  at  5\  per  cent,  for  purchas- 
ing flour  to  the  amount  of  $1365.25  ? 

9.  I  send  my  agent  $1000  to  be  laid  out  in  cotton,  and 
pay  him  5-f  per  cent,  commission  :  what  is  his  commission; 
and  how  many  dollars  worth  of  cotton  shall  I  receive  ? 

10.  A  man  sent  a  broker  $10478.13  to  lay  out  in  stocks 
after  deducting  his  brokerage,  at  %  per  cent. :  what  was 
the  brokerage ;  and  how  much  stock  did  he  receive  ? 

QUEST. — 232.  How  compute  commission,  brokerage,  &c.  on  any 
given  sum  ?  How,  when  the  commission  is  to  be  deducted  in  advance  \ 


ART.  232.]  STOCKS.  213 

11.  A  merchant  negotiated  a  bill  of  exchange  of  $5000 
with  a  broker,  and  agreed  to  give  him  7  per  cent. :  how 
much  did  the  broker  receive  ? 

12.  What  is  the  brokerage  on  $8265,  at  5i  per  ct.? 

13.  What  is  the  brokerage  on  $6524.13,  at  8  per  cent? 

14.  What  is  the  commission  on  $146.356,  at  20  per 
cent.  ? 

15.  What  is  the  commission  on  $1625.75,  at  25  per 
cent.  ? 

16.  What  is  the  commission  on  $25026.10,  at  15  per 
cent.  ? 

17.  What  is  the  brokerage  on  $50265.95,  at  3£  per 
cent.  ? 

18.  What  is  the  brokerage  on  $38212.085,  at  1^  per 
rent.  ? 

19.  What  is  the  brokerage  on  $752600,  at  1  per  cent.  ? 

20.  What  is   the  brokerage  on  $1000000,  at  $  per 
cent.  ? 

21.  Bought  10  shares  of  bank  stock,  for  which  I  agreed 
to  pay  4  per  cent,  advance:   how  much  did  the  stock 
cost  me  ? 

Suggestion. — The  stock  manifestly  cost  me  its  par  value, 
viz  :  $1000,  together  with  4  per  cent,  of  it.  (Art.  229.  Obs.) 
Now  $1000x.04=$40;  and  $1000+$40=$1040.  Ans. 

22.  A  man  bought  5  shares  of  the  Boston  and  Provi- 
dence Railroad  stock,  at  5£  per  cent,  advance  :  what  did 
his  stock  cost  him  ? 

23.  A  stock  broker  bought  15  shares  of  the  New  York 
and  Erie  Railroad  stock  at  par,  and  sold  them  at  15  per 
cent,  discount :  what  did  they  come  to  ? 

24.  Sold  29  shares  in  the  American  Manufacturing  Co., 
at  1 6  per  cent,  advance  :  what  did  they  come  to  ? 

25.  A  stock  jobber  bought  45  shares  of  the  Auburn  and 
Rochester  Railroad  stock  at  3  per  cent,  discount,  which 
he  sold  at  7  per  cent,  advance :  how  much  did  he  make 
by  the  transaction  ? 

26.  A  widow  invested  $9000  in  the  Commonwealth 
Bank  stock  at  par,  and  finally  sold  it  at  75  per  cent,  dis- 
count :  how  much  did  she  lose  ? 


214  INTEREST.  [SECT  IX, 

27.  A  man  owned  53  shares  of  the  Long  Island  Rail- 
road stock,  which  he  sold  at  auction,  at  13  per  cent,  ad- 
vance :  how  much  did  they  come  to  ? 

28.  Bought  38  shares  in  the  Union  Gas  Co.  at  7  per 
cent,  advance,  and  sold  them  at  5  per  cent,  discount :  how 
much  was  my  loss  ? 


INTEREST. 


233*  INTEREST  is  the  sum  paid  for  the  use  of  money 
by  the  borrower  to  the  lender.  It  is  reckoned  at  a  given 
per  cent,  per  annum  ;  that  is,  so  many  dollars  are  paid  for 
the  use  of  $100  for  one  year ;  so  many  cents  for  100  cents : 
so  many  pounds  for  £100  ;  &c. 

OBS.  The  learner  should  be  careful  to  notice  the  distinction  be 
tween  Commission  and  Interest.  The  former  is  reckoned  at  a  certain 
per  cent,  without  regard  to  time;  (Art.  231  ;)  the  latter  is  reckoned  at 
a  certain  per  cent,  for  one  year ;  consequently,  for  longer  or  shorter 
periods  than  one  year,  like  proportions  of  the  percentage  for  one  year 
are  taken. 

The  term  per  ann um}  signifies  for  a  year. 

234*  The  money  lent,  or  that  for  which  interest  is 
paid, is  called  the  principal. 

The  per  cent,  paid  per  annum,  is  called  the  rate. 

The  sum  of  the  principal  and  interest,  is  called  thf 
amount.  Thus,  if  I  borrow  $100  for  1  year,  and  agre* 
to  pay  5  per  cent,  for  the  use  of  it,  at  the  end  of  the  yeai 
I  must  pay  the  lender  the  $100  the  sum  which  I  borrowed 
and  $5  interest,  making  $105.  The  principal  in  thiz 
case,  is  $100  ;  the  interest  $5  ;  the  rate  5  per  cent. ;  and 
the  amount  $105. 


QUEST  —232.  What  is  Interest  ?  How  is  it  reckoned  ?  Ol>s.  Wha 
js  meant  b>  the  term  per  annum  ?  234.  What  is  meant  by  the  princi 
pal  ?  The  rate  ?  The  amount  ?  235.  How  is  the  rate  usually  detea 
mined  ?  Is  it  the  same  everywhere  ? 


w 

ARTS.  233-236.]  INTEREST.  215 

235*  The  rate  of  interest  is  usually  established  by 
aw.  It  varies  in  different  countries  and  in  different  parts 
of  our  own  country. 

OBS.  1.  The  legal  rate  of  interest  in  New  England,  New  Jersey, 
Pennsylvania,  Delaware,  Maryland,  Virginia,  North  Carolina,  Ten- 
nessee, 'Kentucky,  Ohio,  Indiana,  Illinois,  Missouri,  and  Arkansas, 
is  6  per  cent. 

In  New  York,  South  Carolina,  Michigan,  Wisconsin,  and  Iowa, 
it  is  7  per  cent. 

In  Georgia,  Alabama,  Mississippi,  and  Florida,  it  is  8  per  cent. ; 
and  in  Louisiana  but  5  per  cent. 

On  debts  and  judgments  in  favour  of  the  United  States,  interest  is 
computed  at  6  per  cent. 

2.  In  England  and  France  the  legal  rate  is  5  per  cent. ;  in  Ireland^ 
6  per  cent.  In  Italy  about  the  commencement  of  the  13th  century, 
Lt  varied  from  20  to  30  per  cent. 

236.  Any  rate  of  interest  higher  than  the  legal  rate, 
is  called  usury,  and  the  person  exacting  it  is  liable  to  a 
neavy  penalty. 

Any  rate  less  than  the  legal  rate  may  be  taken,  if  the 
parties  concerned  so  agree. 

OBS.  1.  When  no  rate  is  mentioned,  the  rate  established  by  the 
laws  of  the  State  in  which  the  transaction  takes  place,  is  always  un- 
derstood to  be  the  one  intended  by  the  parties. 

2.  The  term  per  annum,  is  seldom  expressed  in  connexion  with  the 
rate  per  cent.,  but  it  is  always  understood ;  for  the  rate  is  the  per  cent. 
paid  per  annum.  (Art.  234.) 

Ex.  1.  What  is  the  interest  of  $15  for  1  year,  at  4  per 
cent.  ? 

Suggestion. — 4  per  cent,  is  -fta  ;  that  is,  $4  for  $100,  4 
cents  for  100  cents,  &c.  (Art.  222.  a.)  Now  as  the  in- 
terest of  $1  (100  cents)  for  a  year,  is  4  cents,  the  interest 
of  $15  for  the  same  time,  is  15  times  as  much.  And  15 
times  4  cents  are  60  cents.  A?is.  60  cents. 

QUEST. — Obs.  What  is  the  legal  rate  of  interest  in  New  England, 
New  Jersey,  &c.  ?  What  is  the  legal  rate  of  interest  in  New  York, 
South  Carolina,  &c.  ?  In  Georgia,  Alabama,  &c.  ?  On  debts  due  the 
United  States  ?  What  is  the  legal  rate  of  interest  in  England  and 
France  ?  Ireland  ?  236.  What  is  any  rate  higher  than  the  legal  rate 
called?  What  is  the  consequence  of  exacting  usury7  Is  it  safe  to 
take  le«*  than  legal  interest?  Obs.  When  no  rate  is  mentioned, 
what  rat*  ia  understood  ? 


216  INTEREST.  [SECT.  IX, 

Operation.        We  multiply  the  principal  by  the  given 

$15  Prin.      rate  per  cent,  expressed  in  decimals,  as  in 

.04  Rate,     percentage;  (Art.  225;)  and  point  off  aa 

$.(50  int         many  decimals  in  the  product  as  there  are 

decimal  places  in  both  factors. 

2.  What  is  the  interest  of  $45  for  1  year,  at  3  per 
cent?     $1.35  Ans. 

3.  What  is  the  interest  of  $32.125  for  1  year,  at  4£ 
per  cent.  ? 

Operation.  4%  per  cent,  expressed  in  deci 

$32. 125  Prin.  mals  is  .045.     (Art.  223.)     Multi- 

.045  Rate.  ply,  &c.  as  above,  and  point  off  6 

160625  decimals  in  the  product.  (Art.  191.) 

128500  '^ne  fractions  of  a  mill  may  be  omit- 

$L445625  Ans.         ted  in  the  answer"     Hence' 

237*  To  find  the  interest  of  any  sum,  at  any  given 
rate  for  1  year. 

Multiply  the  principal  by  the  given  rate  per  cent,  expressea 
in  decimals,  and  point  off  the.  product  as  in  multiplication  of 
decimal  fractions.  (Art.  191.) 

The  amount  is  found  by  adding  the  principal  and  interest 
together.  (Art.  234.) 

OBS.  1.  In  adding  the  principal  and  interest,  care  must  be  taken  to 
add  dollars  to  dollars,  cents  to  cents,  &c.  (Art.  211.) 

2.  When  the  rate  per  cent,  is  lees  than  10,  a  cipher  must  always 
oe  prefixed  to  the  figure  denoting  it.  (Art.  223.  Obs.  1.)  It  is  highly 
important  that  the  principal  and  the  rate  should  both  be  written  cor- 
rectly, in  order  to  prevent  mistakes  in  pointing  off  the  product. 

4.  What  is  the  interest  of  $75.21  for  1  year,  at  6  pe; 
cent?     $4.5126.  Ans. 

5.  What  is  the  interest  of  $100  for  1  year,  at  5  pel 
cent.  ?  at  6  per  cent.  ?  at  4  per  cent.  ?  at  7  per  cent.  ? 

QUEST. — 237.  How  do  you  compute  interest  for  1  year?  How  find 
the  amount  ?  Obs.  What  precaution  is  necessary  in  adding  the  princi 
pal  and  interest  together  ?  When  the  rate  is  less  than  10  per  cent, 
now  is  it  written  ? 


ARTS.  237,  238.1  INTEREST.  217 

6.  What  is  the  interest  of  $35.31  for  1  year,  at  6  per 
cent.  ? 

7.  What  is  the  interest  of  $50.10  for  1  year,  at  7  per 
cent.  ? 

8.  What  is  the  interest  of  $63  for  1  year,  at  5£  per 
cent.  ? 

9.  What  is  the  interest  of  $136.75  for  1  year,  at  4% 
per  cent.  ? 

10.  What  is  the  interest  of  $260.61  for  1  year,  at  6 
per  cent.  ?     What  is  the  amount  ? 

Ans.  $15.636  int.  $276.246  amount. 

11.  What  is  the  interest  of  $140.25  for  1  year,  at  7 
per  cent.  ?     What  is  the  amount  ? 

12.  What  is  the  interest  of  $163.40  for  1  year,  at  8 
per  cent.  ?     What  is  the  amount  ? 

13.  What  is  the  interest  of  $400  for  1  year,  at  6  per 
cent.  ?     What  is  the  amount  ? 

14.  What  is  the  amount  of  $500  for  1  year,  at  7  per 
cent.  ? 

15.  What  is  the  amount  of  $1000  for  1  year,  at  8  per 
cent.  ? 

16.  What  is  the  interest  of  $100  for  3  years,  at  6  pei 
cent,  per  annum  ? 


Operation.  ,  T1Je  interest  for   3  Y^rs  is 

^  .  plainly  3  times  as  much  as  for 

1  year.     We  therefore  first  find 

the  interest  for  1  year  as  above, 

$6.00  Int.  1  y.  which  is  $6  ;  then  multiplying 

3  No.  of  y.  this  by  3,  gives  the  interest  for 

$18JOO  Int.  for  3  y.       3  Years-     Hence, 

238.  To  compute  the  interest  of  any  sum  for  a  given 
number  of  years. 

First  find  the  interest  of  the  given  sum  for  I  year,  at  the 
given  rate;  (Art.  237  ;)  then  multiply  the  interest  of  1  yea/r 
by  the  given  number  of  years. 


^— 238.  How  is  interest  computed  for  any  number  of  years? 


218  INTEREST  [SECT.  IX. 

17.  At  5  per  cent,  per  annum,  what  is  the  interest  ol 
$45  for  4  years?  Ans.  $9. 

18.  At  6  per  cent.,  what  is  the  interest  of  $200  for  5 
years  ?     What  is  the  amount  ? 

19.  At  7  per  cent.,  what  is  the  interest  of  $250  for 
10  years?     What  is  the  amount? 

20.  At  8  per  cent.,  what  is  the  interest  of  $340.50  for 
3  years  ?     What  is  the  amount? 

21.  At  6  per  cent,  per  annum,  what  is  the  interest  of 
$100  for  1  month? 

Operation.  *  month  is  -^  of  12  months 

...  or  a  year,  therefore  the  inter- 

'  est  for  l  month  wil1  be       as 


nr 

much  as  the  interest   for   1 


12)6.00  Int.  for  1  y.  year.     Now   the   interest   of 

$^50  Int.  for  1m.          $100  for   l   7ear  is  $6>  and 
-&  of  $6,  is  50  cts.     In  like 

manner  any  number  of  months  may  be  considered  a  frac- 
tional part  of  a  year,  and  the  interest  for  them  may  be 
computed  in  the  same  way.  Hence, 

239.  To  compute  the  interest  of  any  sum  for  a  given 
number  of  months. 

First  find  the  interest  for  1  year  as  above  ;  then  take  such 
a  fractional  'part  of  1  year's  interest^  as  is  denoted  by  the 
given  number  of  months. 

Thus,  for  1  month  take  -fa  of  1  year's  interest  ;  for  2 
months,  -&  or  £  ;  for  3  months,  -^  or  -J-  ;  for  4  months,  -fa 
or  •£  ;  for  6  months,  -fa  or  %  ;  &c. 

22.  At  5  per  cent.,  what  is  the  interest  of  $600  for  6 
months.  Ans.  $15. 

23.  A*.  7  per  cent.,  what  is  the  interest  of  $250  for  4 
months? 


QUEST.—  239.  How  is  interest  computed  for  months  ?  For  2  months, 
what  part  would  you  take?  For  3  months?  4  months?  5  months  I  6 
months?  7  months?  S  months?  9  months?  10  months?  11  months? 


A.Rrs.  239, 240.]  INTEREST.  219 

24.  What  is  the  interest  of  $375.31  for  3  months,  at  6 
per  cent.  ? 

25.  What  is  the  interest  of  $60  for  7  months,  at  8  per 
cent.  ?     What  is  the  amount  ? 

26.  What  is   the  interest  of  $96  for    10  months,  at  6 
per  cent.  ?     What  is  the  amount  ? 

27.  At  6  per  cent.,  what  is  the  interest  of  $600  for  1 
day? 

Operation.  1  day  is  -gV  of  30  days,  or 

$600  Prin.  ?  m<on1th  J  he™e  th?  int,.erf ' 


1j 
06  Rate.  *or  1  "a     W1^  "Q         °* 


interest  for    1    month.       If, 
I  In.  for  1  y.       therefore,  we  find  the  inter- 
30)3.00  In.  for  1  m.     est  for  j  monthj  and  take  ^ 

Ans.  $0.10  In.  for  1  d.       of  this,  it  will  evidently  be 
the  interest  for    1  day.     In 

like  manner,  any  number  of  days  may  be  considered  a 
fractional  part  of  a  month,  and  the  interest  for  them  may 
be  found  in  the  same  way.  Hence, 

24O.  To  compute  the  interest  of  any  sum  for  a  given 
number  of  days. 

First  find  the  interest  foY  \  month  as  above,  then  take  such 
a  fractional  part  of  1  montn's  interest  as  is  denoted  by  the. 
given  number  of  days.  Thus  for  1  day  take  •£$  of  1  months 
interest  ;  for  2  days,  -g'V,  or  tV  ;  for  3  days.  -^5-,  or  -rV  ;  for 
10  days,  i  ]  for  20  days,  -f  ;  4*c- 

28.  At  4  per  cent.,  what  is  the  interest  of  $470  for  10 
days?  Ans.  $0.522. 

29.  What  is  the  interest  of  $1000  for  1  y.  1m.  and  1 
d.,  at  6  per  cent.  ? 

30.  What  is  the  interest  of  $42.50  for  2  years  and  6 
months,  at  7  per  cent.  ? 

31.  What  is  the  interest  of  $69.46  for  1  year  and  8 
months,  at  8  per  cent.  ? 


QUEST. — 240.  How  is  interest  computed  for  days  ?    For  2  days, 
tthat  part  would  you  take  ?    Far  5  day  ?  7  days  ?  12  days  ?  25  days  * 


220  INTEREST.  [SECT.  IX, 

241.    From  the  foregoing  principles  we  may  deduce 
the  following  general 

RULE  FOR  COMPUTING  INTEREST. 

I.  FOR  ONE  YEAR.     Multiply  the  principal  by  the  given 
"ate,  and  from  the  product  point  off  as  many  figures  for  deci> 
mals,  as  there  are  decimal  places  in  both  factors.  (Art.  237.) 

II.  FOR  TWO  OR  MORE  YEARS.     Multiply  the  interest  oj 
1  year  by  the  given  number  of  years.  (Art.  238.) 

III.  FOR  MONTHS.      Take  such  a  fractional  part  of  1 
year's  interest,  as  is  denoted  by  the  given  number  of  months. 
(Art.  239.) 

IV.  FOR  DAYS.     Take  such  a  fractional  part  of  1  months 

as  is  denoted  by  the  given  number  of  days. 


OBS.  1.  In  calculating  interest,  a  month,  whether  it  contains  30  or 
31  days,  or  even  but  28  or  29,  as  in  the  case  of  February,  is  usually 
assumed  to  be  one  twelfth  of  a  year. 

2.  In  calculating  interest  30  days  are  considered  a  month ;  conse- 
quently the  interest  for  1  day,  or  any  number  of  days  under  30,  is  so 
many  thirtieths  of  a  month's  interest.  (Art.  170.  Obs.  2.) 

This  practice  seems  to  have  been  originally  adopted  on  account  of 
its  convenience.  Though  not  strictly  accurate,  it  is  sanctioned  by 
custom,  and  is  everywhere  allowed  by  law. 

32.  What  is  the  interest  of  $45.23  for  1  year  and  2 
months,  at  5  per  cent.  1 

33.  What  is  the  interest  of  $43.01  for  2£  years,  at  7 
per  cent.  ? 

34.  What  is  the  interest  of  $215.135  for  2  years  and  3 
months,  at  6  per  cent.  ? 

35.  At  8  per  cent.,  what  is  the  interest   of  $75.98  for  3 
years  ? 

36.  At  5£  per  cent.,  what  is  the  interest  of  $939  for  4 
years  ? 

37.  At  6  per  cent.,  what  is  the  interest  of  $137.50  for  6 
months  ? 


QUEST.  —  241.  Wha  :s  the  general  rule  for  computing  interest  ? 
In  reckoning  interest,  y  <at  part  of  a  year  is  a  month  considered  ?    How 


Obs. 
How 
many  days  are  consider*    \  month  t    Is  this  practice  strictly  accurate  ' 


ARTS.  241-244.]  INTEREST.  221 

38.  At  7  per  cent.,  what  is  the  interest  of  $1500  for 
10  days? 

39."  At  20  per  cent,  what  is  the  interest  of  $3000  for 
3  days? 

40.  At  12£  per  cent.,  what  is  the  interest  of  $1736.25 
for  6  months  ? 


SECOND  METHOD  OF  COMPUTING  INTEREST. 

242*  There  is  another  method  of  computing -interest, 
tvhich  is  very  simple  and  convenient  in  its  application, 
particularly  when  the  interest  is  required  for  months  and 
days,  at  6  per  cent. 


INTEREST  OF  SI  FOR  MONTHS. 

243.  We  have  seen,  (Art.  237,)  that, 

For  1  year,    the  interest  of  $1  is                   6  cents,  which  is  $-06, 

"    1  month,            "            "  is  -fa         of  6  cents,    "  "  .005; 

"    2  months,          "             "  is  •&,  or  £  of  6  cents,    "  "  .01; 

"    3  months,          "            "  is  ^,  or  |  of  6  cents,     "  "  .015; 

"    4  months,          «             "  is  -£j-,  or  $  of  6  cents,     "  «  .02; 

"    5  months,          "             "  is  -&         of  6  cents,     "  "  .025; 

"    6  months,          «             »  is  •&,  or  £  of  6  cents,    "  "  .03; 

That  is,  the  interest  of  $1  for  1  month,  at  6  per  cent.,  u 
5  mills ;  and  for  every  2  months,  it  is  1  cent,  fyc.  Hence, 

244*  To  find  the  interest  of  $1  for  any  number  of 
months,  at  6  per  cent. 

Multiply  the  interest  of  $1  for  1  month,  ($.005,)  by  the 
given  number  of  months,  and  point  off"  3  decimal  figures  in 
the  product.  (Art.  191.) 

1.  At  6  per  cent.,  what  is  the  interest  of  $1  for  7 
months  ? 


QUEST.— 244.  How  may  the  interest  of  $1  be  found  for  any  number 
of  months,  at  6  per  c«nt.  t 


222  INTEREST.  [SECT.  IX 

2.  At  6  per  cent.,  what  is  the  interest  of  $1   for  8 
months  ? 

3.  At  6  per  cent.,  what  is  the  interest  of  $1  for  9 
months?     For  10  months?     For  11  months? 

4.  At  6  per  cent.,  what  is  the  interest  of  $1  for  14 
months?     For  15  months?     For  18  months? 

INTEREST  OF  IS  FOR  DAYS. 

245..  Since  the  interest  of  $1  for  1  mo.  (30  d.)  is  5 

mills,  or  $.005,  (Art.  243,) 

For  6  days  (£  of  30  d.)  the  interest  of  Si  is  i  of  5  mills,  or  S-001 
"  12  days  (f  of  30  d.)  «  «  is  -f  of  5  mills,  or    .002 

"  18  days  (f  of  30  d.)  "  <  is  f  of  5  mills,  or     .003 ; 

"  24  days  (-f-  of  30  d.)  «  «  is  f  of  5  mills,  or    .004 ; 

"    3  days  (-fc  of  30  d.)  «  «  is  ^  of  5  mills,  or  .0005 : 

That  is,  the  interest  of  $1  for  every  6  days,  is  1  mill,  or 
$.001 ;  and  for  any  number  of  days,  it  is  as  many  mills, 
or  thousandths  of  a  dollar,  as  6  is  contained  times  in  the 
given  number  of  days.  Hence, 

246.  To  find  the  interest  of  $1  for  any  number  of 
days,  at  6  per  cent. 

Divide  the  given  number  of  days  by  6,  and  set  the  first 
quotient  figure  in  thousandths'  place,  when  the  days  are  6,  or 
more  than  6 ;  but  in  ten  thousandths'  place,  when  they  are 
less  than  6. 

OBS.  For  60  days  (2  mo.)  the  interest  of  SI  is  1  cent;  (Art.  243;) 
in  this  case,  therefore,  the  first  quotient  figure  must  occupy  hun- 
dredth^ place. 

5.  What  is  the  interest  of  $1  for  1  day,  at  6  per  cent, 
expressed  decimally  ?  Ans.  $.000166-f- 

6.  What  is  the  interest  of  $1  for  9  days,  at  6  per  cent.  ? 
22  days?  4  days?   14  days? 


QUEST. — S46.  How  may  the  interest  of  $1  be  found  for  any  number 
of  days,  at  6  per  cent.  ? 


ARTS.  245-247.J  INTEREST.  223 

7.  What  is  the  interest  of  $1  for  10  days,  at  6  per 
cent.?  16  days?  20  days?  24  days?  27 days?  28  days? 

8.  What  is  the  interest  of  $1  for  1  year,  5  months,  and 
3  days,  at  6  per  cent.  ? 

Solution. — For  1  year,  the  interest  of  $1  is  $.06 
"     5  months,         "         "       "     .025 
"     3  days,  "         "       "     .0005 

Ans.  $.0855 

9.  What  is  the  interest  of  81  for  2  years,  7  months, 
and  20  days,  at  6  per  cent.  ? 

10.  What  is  the  interest  of  $1  for  3  years,  1  month, 
and  15  days,  at  6  per  cent.  ? 

11.  What  is  the  interest  of  $145  for  6  months,  and  24 
days,  at  6  per  cent.  ? 

Operation.  Since  the  int.  of  SI  for  1  mo.  is  5 

$145  Prin  mills,  or  1  cent  for  2  mo.,  it  is  manifest 

0  2         "  ,  the  answer  may  be  found  by  multiply- 

as  =  2  the  mo.  ing  the  prin    ty  juif  the  number  of 

435  Int.  for  6  mo.  months,  regarding  the  days  as  a  frac- 

t-c,        <t        24  d  tional  Par^  °f  a  mo-  >  f°r>  tne  int-  °f  SI 

is  equal  to  half  as  many  cents  as  there 

$4.93  Ans.      '  are  months  in  the  given  time. 

247.  From  these  illustrations  we  may  derive  a 
SECOND  RULE  FOR  COMPUTING  INTEREST. 

Multiply  the  principal  by  the  interest  of  $1  for  the  given 
time,  and  point  off  the  product  as  before.  (Art.  241.) 

Or,  multiply  the  principal  by  half  the  number  of  months, 
^ind  point  off  two  more  decimals  in  the  product  than  there 
are  decimal  jigures  in  the  multiplicand. 

OBS.  1.  In  the  latter  method,  the  years  must  be  reduced  to  months, 
and  the  days  to  the  fraction  of  a  month,  then  take  half  of  them. 

The  interest  at  any  other  rate,  greater,  or  less  than  6  per  cent,  may 
be  found  by  adding  to,  or  subtracting  from  the  interest  at  G  per  cent., 
such  a  fractional  part  of  itself,  as  the  required  rate  exceeds  or  falls 
short  of  6  per  cent.  Thus,  if  the  required  rate  is  7  per  cent.,  first 
find  the  interest  at  6  per  cent.,  then  add  •£  of  it  to  itself;  if  5  per 
cent.,  subtract  •£-  of  it  from  itself,  &c. 

QUEST.— 247.  What  is  the  second  method  of  computing  interest! 
0/8.  When  the  rate  is  greater  or  less  than  6  per  cent.,  how  proceed  ? 


\ 

224  INTEREST.  [SECT.  IX. 

2.  When  it  is  required  to  compute  the  interest  on  a  note,  we  must 
first  find  the  time  for  which  the  note  has  been  on  interest,  by  sub- 
tracting the  earlier  from  the  later  date;  (Art.  170;)  then  cast  the  in- 
terest on  the  face  of  the  note  for  the  time,  by  either  of  the  preceding 
methods.  (Arts.  241,  247.) 

13.  What  is  the  interest  of  $300  for  4  months,  and  18 
days,  at  7  per  cent.  ? 

Operation. 
$300  Prin. 
.023  int.  of  $1  for?  The  required  rate  is  1 

the  time.     5  ,  „ 

QQQ  per  cent,  more  than  6  per 

gOO  cent-  j  we  therefore  find  the 

interest  at  6  per  cent,  and 
6)$6.900=Int.  at  6  per  ct.    add  i  rf  it  (0  ^ 

1  150=£  of  6  per  cent. 
Ans.  $&050~  Int.  at  7  per  ct. 

14.  At  5  per  cent.,  what  is  the  interest  of  $256.25  for 
9  months  and  15 days? 

15.  What  is  the  interest  of  $450  from  Jan.  1st,  1844. 
to  March  13th,  1845,  at  6  per  cent.  ? 

Operation.  $450  Principal. 

Yr       mo       £  -072  Int.  of  $1  for  the  time. 

1845  "  3  "   13  "900 

1844  "   1  "     I  IIJ^L 

Time      1  "  2  "   12  $32.400  Ans. 

EXAMPLES   FOR    PRACTICE. 

1.  What  is  the  interest  of  $45.25  for  8  months,  at  6 
per  cent.  ? 

2.  What  is  the  interest  of  $167.375  for  6  months,  at  6 
per  cent  ? 

3.  What  is  the  interest  of  $93.86  for  3  months  and  15 

days,  at  6  per  cent.  ? 

4.  What  is  the  interest  of  $110  for  1  month  and  20 
days,  at  6  per  cent.  ? 

5.  At  7  per  cent.,  what  is  the  interest  of  $158.91   for 
I  year  and  3  months  ? 

QUEST.— 347.  How  compute  the  interest  on  a  nete? 


ART.  247.]  INTEREST.  225 

6.  At  7  per  cent,  what  is  the  amount  of  $217  for  1 
year  and  8  months  ? 

7.  At  6  per  cent.,  what  is  the  amount  of  $348.10  for 
2  years  and  1  month  ? 

8.  At  7  per  cent,  what  is  the  interest  of  $400  for  1 
year  and  6  months  ? 

9.  At  7  per  cent,  what  is  the  amount  of  $213.01  for  9 
months? 

10.  At  5  percent,  what  is  the  amount  of  $603  for  2 
years  and  5  months  ? 

11.  What  is  the  amount  of  $861  for  8  months  and  24 
days,  at  6  per  cent.  ? 

12.  What  is  the  amount  of  $1236  for  3  months  and  14 
days,  at  7  per  cent  ? 

13.  What  is  the  interest  of  $1400  for  1  year,  1  month 
and  9  days,  at  7  per  cent.  ? 

14.  What  is  the  interest  of  $469.20  for  27  days,  at  8 
per  cent.  ? 

15.  What  is  the  amount  of  $705  for  5  years,  at  9  per 
cent.  ? 

16.  What  is  the  amount  of  $1000  for  10  years,  at  5 
per  cent  ? 

17.  What  is  the  amount  of  $1650.06  for  20  years,  at 
7  per  cent.  ? 

18.  What  is  the  amount  of  $2500  for  7  years,  at   15 
per  cent.  ? 

19.  At  4£  per  cent.,  what  is  the  interest  of  $17000  for 
1-J-  years? 

20.  At  7i  per  cent,  what  is  the  interest  of  $1625.81 
for  45  days  ? 

21.  At  121  per  cent,  what  is  the  amount  of  $165.13 
for  33  days  ? 

22.  At  7  per  cent.,  what  is  the  amount  of  $8531  for  63 
days? 

23.  At  6  per  cent,  what  is  the  amount  of  $16021  foi 
93  days  ? 

24.  What  is  the  interest  on  a  note  of  $65,  dated  Jan. 
10th,  1844,  to  May  16th,  1845,  at  6  percent.? 

25.  What  is  the  interest  of  $170  from  June  19th,  1840, 
to  July  1st,  1841,  at  7  per  cent? 


226  INTEREST.  [SECT.  IZ. 

26.  What  is  the  interest  of  $105.63  from  Feb.  22d 
1839,  to  Aug.  10th,  1840,  at  5  per  cent.  ? 

27.  What  is  the  interest  of  $234  from  April   I Oth, 
1834,  to  Oct.  1st,   1835,  at  6  per  cent.? 

28.  What  is  the  interest  of  $195.22  from  June  25th 
1838,  to  March  31st,  1840,  at  6  per  cent.? 

29.  What  is  the  interest  of  $391  from  Sept.  1st,  1840 
to  Nov.  30th,  1841,  at  8  per  cent.? 

30.  What  is  the  interest  of  $510.83  from  March  21st, 
1842,  to  Dec.  30th,  1842,  at  7  per  cent? 

31.  At  6  per  cent,  what  is   the  interest  of  $469.65 
from  August  10th,  1843,  to  Feb.  6th,  1844  ? 

32.  At  7  per  cent.,  what  is  the  amount  due  on  a  note 
of  $285,  dated  March  15th,  1844,  and  payable  Sept.  18th 
1845? 

33.  At  6  per  cent.,  what  is  the  amount  due  on  a  note 
of  $S9l,  dated  Oct.  9th,   1844,  and  payable  March  1st, 
1845? 

34.  At  5  per  cent.,  what  is  the  amount  of  $623  from 
Feb.   19th,  1844,  to  Aug.  10th,  1844? 

35.  At  4  per  cent.,  what  is  the  amount  of  $589.20  from 
January  10th,   1844,  to  January  13th,  1845? 

36.  At  4  per  cent,  what  is  the  amount  of  $731.27  from 
July  1st,  1844,  to  April  4th,  1845? 

37.  What  is  the  interest  of  $849  from  July  4th,  1841, 
to  July  7th,  1845,  at  6  per  cent.  ? 

38.  What  is  the  interest  of  $966  from  Jan.  1st,  1842, 
to  March  20th,   1844,  at  7  per  cent.  ? 

39.  What  is  the  interest  of  $1539  from  May  21st,  1842 
to  Aug.  19th,  1843,  at  6  per  cent  ? 

40.  What  is  the  amount  of  $  1 100  from  June  1 5th,l  840, 
to  Aug.  3d,  1845,  at  5  per  cent.  ? 

41.  What  is  the  amount  of  $1  for  50  years,  at  6  per  ct.  ? 
At  7  per  cent.  ? 

42.  What  is  the  amount  of  one  cent  for  500  years,al 
7  per  cent  1 


A.RT.    248.]  INTEREST.  227 

PARTIAL    PAYMENTS. 

248*  WThen  partial  payments  are  made  and  endorsed 
upon  Notes  and  Bonds,  the  rule  for  computing  the  inter- 
est adopted  by  the  Supreme  Court  of  the  United  States,  is 
the  following. 

1  "  The  rule  for  casting  interest,  when  partial  payments 
have  been  made,  is  to  apply  the  payment,  in  the  first  place,  to 
the  discharge  of  the  interest  then  due. 

II.  ^  If  the  payment  exceeds  the.  interest,  the  surplus  goes 
towards  discharging  the  principal,  and  the  subsequent  interest 
is  to  be  computed  on  the  balance  of  principal  remaining  dm. 

III.  "If  the  payment  be  less  than  the  interest,  the  surplus 
of  interest  must  not  be  tdken  to  augment  the  principal ;  but 
interest  continues  on  the  former  principal  until  the  period 
when  the  payments,  taken  together,  exceed  the  interest  due, 
and  then  the  surplus  is  to  be  applied  tc  wards  discharging  the 
principal ;  and  interest  is  to  be  computed  on  the  balance  as 
aforesaid." 

Note. — The  above  rule  is  adopted  by  Massachusetts,  New  York,  and 
the  other  States  of  the  Union,  with  but  few  exceptions.  It  is  given 
in  the  language  of  the  distinguished  Chancellor  Kent. — Johnson's 
Chancery  Reports,  Vol.  I.  p.  17. 


$850.  NEW  HAVEN,  Jan.   1st,   1841. 

43.  For  value  received,  I  promise  to  pay  George  How- 
land,  or  order,  eight  hundred  and  fifty  dollars,  on  demand, 
with  interest  at  6  per  cent. 

JOHN  HAMILTON. 

The  following  payments  were  endorsed  on  this  note 

July  1st,  1841,  received  $100.62. 
Dec.  1st,  1841,  received  $15.28. 
Aug.  13th,  1842,  received  $175.75. 

What  was  due  on  taking  up  the  note,  Jan.  1st,  1843  1 

QUEST.— 248.  What  is  the  general  method  of  casting  interest  oa 
Motes  laid  Bonds,  when  partial  payments  have  been  made  1 


228 


INTEREST. 


[SECT. 


Operation. 
Principal, 

Interest  to  first  payment,  July  1st,  (6  months,) 
Amount  due  on  note  July  1st,     - 
1st  payment,  (to  be  deducted  from  amount,) 

Balance  due  July  1st, 

Int.  on  Bal.  to  2d  pay't  Dec.  1st,  (5  mo.,)  $19.37 

2d  pay't  (which  is  less  than  the  inter- 

est  then  due,) 

Surplus  interest  unpaid  Dec.  1st, 
Int.  continued  on  Bal.  from  Dec.  1st, 

1842,  to  Aug.  13th,  (8  mo.,  12  d.,) 
Amount  due  Aug.   13th,  1842. 
3d  payment  (being  greater  than  the  interest 

now  due)  is  to  be  deducted  from  the  am't. 
Balance  due  Aug.  13th, 
Int.  on  Bal.  to  Jan.  1st,  (4  mo.,  18d.,) 

Bal.  due  on  taking  up  the  note,  Jan.  1st,  1843,    $650.38 


15.28 
$409" 

32.54 


$850.00 
25.50 

$875.50 
100.62 

$774.88 


36.63 


$811.51 

175.75 

$635.76 
14.62 


$500. 


NEW  YORK,  May  10th,  1842. 


44.  For  value  received,  I  promise  to  pay  James  Mon- 
roe, or  order,  five  hundred  dollars  on  demand,  with  in- 
terest at  7  per  cent 

HENRY  SMITH. 

The  following  sums  were  endorsed  upon  it : 

Received,  Nov.  10th,  1842,  $75. 
Received,  March  22d,  1843,  $100. 

What  was  due  on  taking  up  the  note,  Sept.  28th,  1843 

$692.35.  BOSTON,  Aug.  15th,  1843. 

45.  Three  months  after  date,  I  promise  to  pay  John 
Warren,  or  order,  six  hundred  and  ninety-two  dollars  and 
thirty-five  cents,  with  interest  at  6  per  cent.,  value  re- 
ceived, SAMUEL  JOHNSON. 


A.RTS.  249, 249.  a.\         INTEREST.  229 

Endorsed,  Nov.  15th,  1843,  $250.375. 
"         March  1st,  1844,     $65.625. 

How  much  was  due  July  4th,  1845  ? 


$1000.  PHILADELPHIA,  June  20th,  1841. 

46.  Six  months  after  date,  I  promise  to  pay  Messrs. 
Carey,  Hart  &  Co.,  or  order,  one  thousand  dollars,  with 
interest,  at  5  per  cent.,  value  received. 

HORACE  PRESTON. 

Endorsed,  Jan.  10th,  1844,  $125. 

"         June  16th,  1844,     $93. 

Feb.  20th,  1845,  $200. 

What  was  the  balance  due  Aug.  1st,  1845  ? 

CONNECTICUT  RULE. 

xJ49»  "Compute  the  interest  on  the  principal  to  the  time  of 
me  first  payment;  if  that  be  one  year  or  more  from  the  time  the  in- 
terest commenced,  add  it  to  the  principal,  and  deduct  the  payment 
from  the  sum  total.  If  there  be  after  payments  made,  compute  the 
interest  on  the  balance  due  to  the  next  payment,  and  then  deduct  the 
payment  as  above ;  and  in  like  manner,  from  one  payment  to  an- 
other, till  all  the  payments  are  absorbed ;  provided  the  time  between 
one  payment  and  another  be  one  year  or  more.  But  if  any  payments 
be  made  before  one  year's  interest  hath  accrued,  then  compute  the 
interest  on  the  principal  sum  due  on  the  obligation,  for  one  year,  add 
it  to  the  principal,  and  compute  the  interest  on  the  sum  paid,  from  the 
time  it  was  paid  up  to  the  end  of  the  year;  add  it  to  the  sum  paid, 
and  deduct  that  sum  from  the  principal  and  interest  added  as  above.* 

"  If  any  payments  be  made  of  a  less  sum  than  the  interest  arisen 
at  the  time  of  such  payment,  no  interest  is  to  be  computed,  but  only 
on  the  principal  sum  for  any  period." — Kirby's  Reports. 

THIRD  RULE. 

24:9.  a.  First  find  the  amount  of  the  given  principal  for  the 
whole  time ;  then  find  the  amount  of  each  of  the  several  payments 
from  the  time  it  was  endorsed  to  the  time  of  settlement.  Finally, 
subtract  the  amount  of  the  several  payments  from  the  amount  of  the 
principal,  and  the  remainder  will  be  the  sum  due. 

*  If  a  year  does  not  extend  beyond  the  time  of  payment;  but  if  it  does,  then 
find  the  amount  of  the  principal  remaining  unpaid,  up  to  the  time  of  settlement, 
likewise  the  amount  of  the  endorsements  from  the  time  they  were  paid  \£  the 
lime  of  settlement,  and  deduct  the  sum  of  these  several  amounts  from  the 
•mount  of  the  principal. 


230  INTEREST.  [SECT.  IX, 

Note.  —  It  will  be  an  excellent  exercise  for  the  pupil  to  cast  the  in- 
terest on  each  of  the  preceding  notes  by  each  of  the  above  rules. 

47.  What  is  the  interest  of  £175,  10s.  6d.  for  1  year,  al 
5  per  cent.  ? 

Operation.  We  first  reduce  the  10s.  6d. 

£175.525  Prin.  to  the  decimal  of  a  pound. 

.05  Rate.  (Art.  200,)  then  multiply  the 

£877625  Int.  for  1  yr.  Principal  by  the  rate  and  point 

2Q  on  the  product  as  in  Art.  241. 

The  fiure  8  on  the  left  of  the 


decimal  point  is  pounds,  and 
those  on  the  right  are  decimals 

d.    6.30000  of  a  pound,  and  must  be  re- 

4  duced  to  shillings,  pence,  and 

far.  1.20000  farthings.  (Art.  201.) 

Ans.  £8,  15s.  6|d.      Hence, 

25O«  To  compute  the  interest  on  pounds,  shillings, 
&c. 

Reduce,  the  given  shillings,  pence,  and  farthings  to  the  de- 
cimal of  a  pound;  (Art.  200  ;)  then  find  the  interest  as  cm 
dollars  and  cents  ;  finally,  reduce  the  decimal  figures  in  the 
answer  to  shillings,  pence,  and  farthings.  (Art.  201.) 

48.  What  is  the  interest  of  £56,  15s.  for  one  year  and 
5  months,  at  6  per  cent.  ?  Ans.  £5,  2s.  If  d. 

49.  What  is  the  interest  of  £75,  12s.  6d.  for  1  year  and 
3  months,  at  7  per  cent.  ? 

50.  What  is  the  interest  of  £96,  18s.  for  2  years  and  6 
months,  at  4-£  per  cent.  ? 

51.  What  is  the  amount  of  £100  for  2  years  and  4 
months,  at  5  per  cent.  'I 

52.  What  is  the  amount  of  £430,  16s.  lOd.  for  1  year 
and  5  months,  at  6  per  cent.  ? 


QUEST. — 250.  How  is  interest  computed  on  pounds,  shillings,  &t  ? 


ARTS.  250-252.]  INTEREST.  231 


PROBLEMS  IN  INTEREST. 

251,  It  will  be  observed  that  there  are  four  'parts  or 
terms  connected  with  each  of  the  preceding  operations, 
viz  :  the  principal,  the  rate  per  cent.,  the  time,  and  the  inter- 
est, or  the  amount.  These  parts  or  terms  have  such  a  re- 
lation to  each  other,  that  if  any  three  of  them  are  given, 
the  other  may  be  fonnd.  The  questions,  therefore,  which 
may  arise  in  interest,  are  numerous ;  but  they  may  be 
reduced  to  a  few  general  principles,  or  Problems. 

OBS.  I.  The  term  Problem,  in  its  common  acceptation,  means  a 
question  proposed,  which  requires  a  solution. 

2.  A  number  or  quantity  is  said  to  he  given,  when  its  value  is  stat- 
ed, or  may  be  easily  infened  from  the  conditions  of  the  question  under 
consideration.  Thus,  when  the  principal  and  interest  are  known, 
the  amount  may  be  said  to  be  given,  because  it  is  merely  the  sum  of 
the  principal  and  interest.  So,  if  the  principal  and  the  amount  are 
known,  the  interest  may  be  said  to  be  givsn,  because  it  is  the  differ- 
ence between  the  amount  and  the  principal. 

252*  To  find  the  interest  on  any  given  sum,  as  in  the 
"oregoing  examples,  the  principal,  the  rate  per  cent.,  and 
the  time  are  always  given.  This  is  the  First  and  most 
important  Problem  in  interest.  The  other  Probkms  will 
now  be  illustrated. 


PROBLEM    II.* 

To  find  the  RATE  PER  CENT.,  the  principal,  the  interest, 
and  the  time  being  given. 

1.  A  man  loaned  $75  to  one  of  his  neighbors  for  4 
years,  and  received  $24  interest :  what  was  the  rate  per 
cent.  1 


QUEST.— 251.  How  many  terms  are  connected  with  each  of  :he  pre- 
ceding examples  ?  What  are  they  ?  Are  they  all  given  ?  When  three 
are  given,  can  the  fourth  be  found  ?  Obs.  What  is  a  problem  ?  When 
is  a  number  or  quantity  said  to  be  given  ?  252.  What  terms  are  given 
when  it  is  required  to  find  the  interest  ? 

*  Should  this  and  the  following  Problems  bo  deemed  too  difficult  for  beginners 
they  can  be  omitted  till  review. 


232  INTEREST.  [SECT.  IX. 

Analysis. — The  interest  of  $75  at  1  per  cent,  for  1  year, 
is  $.75,  and  for  4  years  it  is  $.75x4=$3.  (Art.  238.) 
Now  since  $3  is  1  per  cent,  interest  on  the  principal  for 
the  given  time,  $24  must  be  -\4-  of  1  per  cent.,  which  is 
equal  to  8  per  cent.  (Art.  121.) 

Or,  we  may  reason  thus :  If  $3  is  1  per  cent,  on  tho 
principal  for  the  given  time,  $24  must  be  as  many  per 
cent,  as  $3  is  contained  times  in  $24 ;  and  $24~$3^8. 

Ans.  8  per  cent. 

PROOF. — $75X-08=$6.00,  the  interest  for  1  year  at 
8  per  cent.,  and  $6x4=$24,  the  interest  of  $75  for  4 
years  at  8  per  cent.  Hence, 

253.  To  find  the  rate  per  cent,  when  the  principal, 
interest,  and  time  are  given. 

First  find  the  interest  of  the  principal  at  1  per  cent  for 
the  given  time ;  then  make  the  interest  thus  found  the  denom- 
inator and  the  given  interest  the  numerator  of  a  common 
fraction,  which  being  reduced,  to  a  whole  or  mixed  number,  will 
give  the  required  per  cent.  (Art.  121.) 

Or,  simply  divide  the  given  interest  by  the  interest  of  the 
principal  at  1  per  cent,  for  the  given  time,  and  the  quotient 
will  be  the  per  cent. 

2.  If  I  borrow  $300  for  2  years,  and  pay  $42  interest, 
what  rate  per  cent,  do  I  pay  ? 

Operation.  The  interest  of  $300  for  2  yrs. 

$6)$42  at  1  per  cent,  is  $6.  (Art.  238.) 

7  Ans.  7  per  ct. 
PROOF.— $300x.07x2=$42. 

3.  If  I  borrow  $460  for  3  years,  and  pay  $82.80  in- 
terest,  what  is  the  rate  per  cent.  ? 

4.  A  man  loaned  $500  for  8  months,  and  received  $40 
interest :  what  was  the  rate  per  cent.  ? 

5.  At  what  rate  per  cent,  must  $450  be  loaned,  to  gain 
$56.50  interest  in  1  year  and  6  months? 

QUEST. — 253.  When  the  principal,  interest,  and  time  are  giTen,  hew 
is  the  rate  per  cent,  found  ? 


V. 


ART.  253.]  INTEREST.  233 

6.  At  what  per  cent,  must  $750  be  loaned,  to  gain 
$225  in  4  years  ? 

7.  A  man  has  $8000  which  he  wishes  to  loan  for  $60C 
per  annum  for  his  support :  at  what  per  cent,  must  he 
loan  it? 

8.  A  gentleman  deposited  $1250  in  a  savings  bank, 
for  which  he  received  $31.25  every  6  months  ;  what  per 
cent,  interest  did  he  receive  on  his  money  ? 

9.  A  capitalist  invested   $9260  in  Railroad  stock,  and 
drew  a  semi-annual  dividend  of  $416.70  :  what  rate  per 
cent,  interest  did  he  receive  on  his  money  ? 

10.  A  man  built  a  hotel  at  an  expense  of  $175000, 
and  rented  it  for  $8750  per  annum  :  what  per  cent,  inter- 
est did  his  money  yield  him  ? 

PROBLEM   III. 

To  find  the  PRINCIPAL,  the  interest,  the  rate  per  cent.,  and 
the.  time  being  given. 

11.  What  sum  must  be  put  at  interest,  at  6  per  cent, 
to  gain  $30  in  two  years? 

Analysis. — The  interest  of  $1  for  2  years  at  6  per  cent., 
(the  given  time  and  rate,)  is  12  cents.  Now  1 2  cents  interest 
is  -iW  of  its  principal  $1  ;  consequently,  $30  the  given 
interest,  must  be  -^  of  the  principal  required.  The 
question  therefore  resolves  itself  into  this :  $30  is  -^  of 
what  number  of  dollars?  If  $30  is  -fW,  liir  is  iV  of 
$30.  which  is  $2i;  and  -HH}=$2ixlOO,  which  is  $250, 
the  principal  required. 

Or,  we  may  reason  thus:  Since  12  cents  is  the  interest 
of  1  dollar  for  the  given  time  and  rate,  30  dollars  must  be 
the  interest  of  as  many  dollars  for  the  same  time  and  rate, 
as  12  cents  is  contained  times  in  30  dollars.  And 
$30-H.12=250.  Am.  $250. 

PROOF.—  $250x06=$! 5.00,  the  interest  for  1  year  at 
the  given  per  cent.,  and  $!5x2=$30,  the  given  interest, 
Hence. 


234  INTEREST.  [SECT.  IX, 

254.  To  find  the  principal,  when  the  interest,  rate 
per  cent,  and  time  are  given. 

Make  the  interest  of  $1  for  the  given  time  and  rate,  the  nu- 
merator, and  100  the  denominator  of  a  common  fraction ;  then 
divide  the  given  interest  by  this  fraction  ;  and  the  quotient 
will  be  tJie  principal  required.  (Art.  141.) 

Or,  simply  divide  the  given  interest  by  the  interest  of  $  1 
for  the  given  time  and  rate,  expressed  in  decimals ;  and  the 
quotient  will  be  the  principal 

12.  What  sum  put  at  interest  will  produce  $13.30  in 
6  months,  at  7  per  cent.  1 

Operation.  The   int.   of  $1   for  6 

$.035)$13.300  mo.  at  7  per  cent,  is  $.035 

"380.   Ans.  $380.        (Art  239') 

13.  A  father  bequeaths  his   son  $500  a  year:  what 
sum  must  be  invested,  at  5  per  cent,  interest,  to  produce 
it? 

14.  What  sum  must  be  put  at  6  per  cent,  interest,  to 
gain  $350  interest  semi-annually  ? 

15.  A  gentleman  retiring  from  business,  loaned  his 
money  at  7  per  cent.,  and  received  $1200  interest  a  year  • 
how  much  was  he  worth  ? 

PROBLEM    IV. 

To  find  the  TIME,  the  principal,  the  interest,  and  the  rate 
per  cent,  being  given. 

16.  A  man  loaned  $80  at  5  per  cent.,  and  received  $10 
interest :  how  long  was  it  loaned  ? 

Analysis. — The  interest  of  $80  at  5  per  cent,  for  1  year 
is  $4.  (Art.  237.)  Now,  since  $4  interest  requires  the 
principal  1  year  at  the  given  per  cent,  $10  interest  will 
require  the  same  principal  -^of  1  year,  which  is  equal  to 
2i  years.  (Art  121.) 


QUEST.— 254.  When  the  interest,  rate  per  cent.,  and  time  are  given, 
how  is  the  principal  found  ? 


ARTS.  254,  255.] 


INTEREST. 


235 


Or,  we  may  reason  thus :  If  $4  interest  requires  the 
use  of  the  given  principal  1  year,  $10  interest  will  re- 
quire the  same  principal  as  many  years  as  $4  is  contained 
times  in  $10.  And  $10-=-$4=2.5.  Ans.  2.5  years.  Hence, 

255*  To  find  the  time  when  the  principal,  interest, 
and  rate  per  cent,  are  given. 

Make  the  given  interest  the  numerator,  and  the  interest  of 
the  principal  for  1  year  at  the  given  rate  the  denominator  of 
a  common  fraction,  which  being  reduced  to  a  whole  or  mixed 
number,  will  give  the  time  required. 

Or,  simply  divide  the  given  interest  by  the  interest  of  the 
principal  at  the  given  rate  for  1  year,  and  the  quotient  will 
be  the  time. 

OBS.  If  the  quotient  contains  a  decimal  of  a  year,  it  should  be  re- 
duced to  months  and  days.  (Art.  201.) 


17.  How  long  will  it  take 


at  5  per  cent,  to 


double  itself;  that  is,  to  gain  $100  interest? 

Operation.         The  interest  of  $100  for   1  year,  at  5  pel 
cent.,  is  $5.  (Art.  237.) 


20  Ans.  20  years. 
PROOF.— $100x.05x20=$100.  (Art.  238.) 

TABLE. 

Showing  in.  what  time  any  given  principal  will  double  itself  at  any  rate, 
*  from  1  to  20  per  cent.  Simple  Interest. 


Percent. 

Years. 

Per  cent. 

Years. 

Per  cent. 

Years. 

Per  cent. 

Years. 

1 

100 

6 

16-f 

11 

9-rV 

16 

6i 

2 

50 

7 

14f 

12 

8-£ 

17 

«H4 

3 

33-3 

8 

12"2 

13 

7^ 

18 

54 

4 

25 

9 

IH 

14 

7-? 

19 

5 

20 

10 

10 

15 

6* 

20     )  5 

QUEST  — 255.  When  the  principal,  interest,  and  rate  per  cent,  are 
given,  how  is  the  time  found?  Obs.  When  the  quotient  contains  a 'de- 
cimal of  a  yeai ,  what  should  be  done  with  it  ? 


236  COMPOUND.  [SECT.  IX, 

18.  In  what  time  will  $500,  at  6  per  cent.,  produce 
$100  interest? 

19.  How  long  will  it  take  $100,  at  6  per  cent.,  to 
double  itself? 

20.  How  long  will  it  take  $100,  at  7  per  cent.,  to  double 
itself? 

21.  How  long  will  it  take  $7250,  at  10  per  cent.,  to 
double  itself? 


COMPOUND    INTEREST. 


S56.  Compound  Interest  is  the  interest  arising  not 
only  from  the  principal,  but  also  from  the  interest  itself. 
after  it  becomes  due. 


OBS.  1.  Compound  Interest  is  often  called  interest  upon  interest. 
2.  When  the  interest  is  paid  on  the  principal  only,  it  is  called  Sim- 
vie  Interest. 

Ex.   1.  What  is  the  compound  interest  of  $500  for  3 
years,  at  6  per  cent.  ? 

Operation. 

$500  principal. 
$500x.06=$  30  Int.  for  1st  year. 


530  Amt.  for  1  year. 
$530x.06=      31.80  Int.  for  2d  year. 

561.80  Amt.  for  2  years. 
$561.80x06=     33.70  Int.  for  3d  year. 
$595.50  Amt.  for  3  years. 

500.00  Prin.  deducted. 
Ans.  $95.50  compound  Int.  for  3  years. 


'QuEST. — 256.  From  what  does  compound  interest  arise  ?     O6.« 
What  is  compound  interest  often  called  ?    What  is  Simple  Interest  I 


ARTS.  256, 257.]  INTEREST.  237 

257*  Hence,  to  calculate  compound  interest. 

Cast  the  interest  on  tlie  given  principal  for  1  year,  or  the 
specified  time,  and  add  it  to  the  principal ;  then  cast  the  inter- 
est on  this  amount  for  the  next  year,  or  specified  time,  and  add 
it  to  the  principal  as  before.  Proceed  in  this  manner  with 
each  successive  year  of  tlie,  proposed  time.  Finally,  subtract 
the  given  principal  from  the  last  amount,  and  the  remainder 
will  be  the  compound  interest. 

2.  What  is  the  compound  interest  of  $350  for  4  years, 
at  6  per  cent.  ? 

3.  What  is  the  compound  interest  of  $865  for  5  years, 
at  7  per  cent.  ? 

4.  What  is  the  amount  of  $250  for  6  years,  at  5  per 
cent,  compound  interest  ? 

5.  What  is  the  amount  of  $1000  for  3  years,  at  4  per 
cent,  compound  interest,  payable  semi-annually  ? 

6.  What  is  the  amount  of  $1200  for  2  years,  at  6  per 
cent,  compound  interest,  payable  quarterly  ? 

7.  What  is  the  amount  of  $800  for  3  years,  at  5  pel 
cent,  compound  interest,  payable  semi-annually  ? 

8.  What  is  the  amount  of  $1500  for  5  years,  at  7  per 
cent,  compound  interest  ? 

9.  What  is  the  amount  of  $2000  for  2  years,  at  3  per 
cent,  compound  interest,  payable  quarterly? 

10.  What  is  the  amount  of  $3500  for  6  years,  at  6  pel 
cent,  compound  interest? 

Note.—  This  and  the  next  two  examples  may  be  solved  either  by 
the  rule,  or  by  the  Table  below. 

11.  What  is  the  amount  of  $1860  for  8  years,  at  7 
per  cent,  compound  interest  ? 

12.  What  is  the  amount  of  $20000  for  10  years,  at  3 
per  cent,  compound  interest  ? 


QUEST.—  257.  How  is  compound  interest  calculated  t 


238 


INTEREST. 


[SECT.  IX. 


TABLE, 

Showing  the  amount  of  SI,  or  £1,  at  3,  4,  5,  C,  and  7  per  cent*, 
mn        pound  interest,  for  any  number  of  years,  from  I  to  35. 


Yrs.  |  3  per  cent.  \  4  per  cent.  \  5  per  cent,  \  6  per  cent.  \  7  percent.\ 


1. 

1.030,000 

1.040,000 

1.050,000 

1.060,000 

1.07,000 

I  2- 

1.060,900 

1.081,600 

1.102.500 

1.123,600 

1.14.490 

3. 

1.092,727 

1.124,864 

1.157,625 

1.191,016 

1.22J504 

4. 

1.125,509 

1.169,859 

1.215,506 

1.262,477 

1.31,079 

5. 

1.159,274 

1.216,653 

1.276,282 

1.338,226 

1.40,255 

6. 

1.194,052 

1.265,319 

1.340,096 

1.418,519 

1.50,073 

7. 

1.229,874 

1.315,932 

1.407,100 

1.503,630 

1.60,578 

8. 

1.266,770 

1.368,569 

1.477,455 

1.593,848 

1.71,818 

9. 

1.304,773 

1.423,312 

1.551,328 

1.689,479 

1.83,845 

10. 

1.343,916 

1.480,244 

1.628,895 

1.790,848 

1.96,715 

11. 

1.384,234 

1.539,454 

1.710,339 

1.898,299 

2.10,485 

12. 

1.425,761 

1.601,032 

1.795,856 

2.012,196 

2.25,219 

13. 

1.468,534 

1.665,074 

1.885,649 

2.132,928 

2.40,984 

14. 

1.512,590 

1.731,676 

1.979.932 

2.260,904 

2.57,853 

15. 

1.557,967 

1.800,944 

2.078^28 

2.396,558 

2.75,903 

16. 

1.604,706 

1.872,981 

2.182,875 

2.540,352 

2.95,216 

17. 

1.662,848 

1.947,900 

2.292,018 

2.692,773 

3.15,881 

18. 

1.702,433 

2.025,817 

2.406,619 

2.854,339 

3.37,293 

19. 

1.753.506 

2.108,849 

2.526,950 

3.025,600 

3.61,652 

20. 

1.806,111 

2.191,123 

2.653,298 

3.207,135 

3.86,968 

21. 

1.860,295 

2.278,768 

2.785,963 

3.399,564 

4.14,056 

i  22. 

1.916,103 

2.369,919 

2.925,261 

3.603,537 

4.43,040 

23. 

1.973.587 

2.464,716 

3.071,524 

3.819,750 

4.74,052 

24. 

2.032J94 

2.563,304 

3.225,100 

4.048,935 

5.07,236 

25. 

2.093,778 

2.665,836 

3.386,355 

4.291,871 

5.42,743 

26. 

2.156,592 

2.772,470 

3.555,673 

4.549,383 

5.80,735 

27. 

2.221,289 

2.883,369 

3.733,456 

4.822,346 

6.21  ,386 

28. 

2.287,928 

2.998,703 

3.920,129 

5.111,687 

6.64,883 

29. 

2.356,566 

3.118,651 

4.116,136 

5.418,388 

7.11,425 

30. 

2.427,262 

3.243,398 

4.321,942 

5.743,491 

7.61,225 

31. 

2.500,080 

3.373,133 

4.538,039 

6.088,101 

8.14,571 

32. 

2.575,083 

3.508,059 

4.764,941 

6.453,386 

8.71,527 

33. 

2.652,335 

3.648,381 

5.003,189 

6.840.590 

9.32,533 

34. 

2.731,905 

3.794,316 

5.253,348 

7.251  ;025 

997,811 

35. 

2.813,862 

3.946,089      5.516,015 

7.686,087 

10.6,765 

ARTS.  258, 259.]  DISCOUNT.  239 

258.  To  calculate  compound  interest  by  the  preced- 
ing Table. 

Find  the  amount  of  $1,  or  £1  for  the  given  number  of 
years  by  the  table,  multiply  it  by  the  given  principal,  and 
the  product  will  be  the  amount  required. 

Subtract  the  principal  from  the  amount  thus  found,  and 
the  remainder  will  be  the  compound  interest. 

1 3.  What  is  the  compound  interest  of  $200  for  1 0  years, 
At  6  per  cent?    What  is  the  amount  ? 

Operation. 

$1.790848  Amt.  of  $1  for  10  years  by  table. 
200  the  given  principal. 

$358.169600  amount  required. 
$200  principal  to  be  subtracted. 
Ans.  $158.1696  interest  required. 

14.  What  is  the  amount  of  $350  for  12  years,  at  4  per 
cent.  ? 

15.  What  is  the  amount  of  $469  for  15  years,  at  3  per 
cent.  ?     What  the  interest  ? 

16.  What  is  the  interest  of  $500  for  24  years,  at  6  per 
cent.  ? 

17.  What  is  the  interest  of  $650  for  30  years,  at  7  per 
cent.  ? 

DISCOUNT. 

259.  DISCOUNT  is  the  abatement  or  deduction  made 
for  the  payment  of  money  before  it  is  due.     For  example, 
if  I  owe  a  man  $100,  payable  in  one  year  without  interest, 
the  present  worth  of  the  note  is  less  than  $100 ;  for,  if  $100 
were  put  at  interest  for  1  year,  at  6  per  cent.,  it  woul-l 
amount  to  $106  ;  at  7  per  cent.,  to  SI 07;  &c.     In  con- 
sideration, there  fore,  of  the  present  'payment  of  the  note,  jus- 
tice requires  that  he  should  make  some  abatement  from  it 
This  abatement  is  called  Discount. 

QUEST. — 258.  How  is  compound  interest  computed  by  the  Table  ? 
259.  What  is  discount  1  What  is  the  present  worth  of  a  debt,  payable 
8,1  some  future  time,  without  interest? 


240  DISCOUNT.  [SECT.  IX 

The  present  worth  of  a  debt  payable  at  some  future  time 
without  interest,  is  that  sum  which,  being  put  at  legal 
interest,  mil  amount  to  the  debt,  at  the  time  it  becomes  due. 

Ex.  1.  What  is  the  present  worth  of  $545,  payable  in 
1  year  and  6  months  without  interest,  when  money  is 
worth  6  per  cent,  per  annum  ? 

Analysis. — The  amount,  we  have  seen,  is  the  sum  o1 
the  principal  and  interest.  (Art.  234.)  Now  the  amount 
of  $1  for  1  year  and  6  months,  at  6  per  cent,  is  $1.09 ; 
(Art.  237  ;)  that  is,  the  amount  is  -ffl-g-  of  the  principal  $1. 
The  question  then  resolves  itself  into  this :  $545  is  -}-£$ 
of  what  principal  ?  If  $545  is  -H-f,  -^  is  545-*- 109,  01 
$5;  and  -HHM&5xlOO,  which  is  $500. 

Or,  we  may  reason  thus:  Since  $1.09  (amount)  requires 
$1  principal  for  the  given  time,  $545  (amount)  will  re- 
quire  as  many  dollars  as  $1.09  is  contained  times  in 
$545;  and  $545-*-$1.09=$500.  That  is,  the  present 
worth  of  $545,  payable  in  1  year  and  6  months,  is  $500, 
which  is  the  answer  required. 

PIIOOF. — $500x.09=$45,  the  interest  for  1  year  and 
6  months;  and  $500+$45=$545  the  given  amount. 
(Art.  247.)  Hence, 

26 O.  To  find  the  present  worth  of  any  sum,  payable 
at  a  future  time  without  interest. 

First  find  the  amount  of  $  1  for  the  time,  at  the  given 
rate,  as  in  simple  interest ;  (Art.  247 ;)  then  divide  the, 
given  sum  by  this  amount,  and  the  quotient  will  be  the  pre- 
sent worth. 

The  present  worth  subtracted  from  the  debt,  will  give  thi 
true  discount. 

OBS.  This  process  is  often  classed  among  the  Problems  of  Interest 
in  which  the  amount,  (which  answers  to  the  given  sum  or  debt,)  the 
rate  per  cent.,  and  the  time  are  given,  to  find  the  principal,  which 
answers  to  the  present  worth. 


QUEST. — 260.  How  do  you  find  the  present  worth  of  a  debt  ?    Hov» 
find  the  discount  \ 


ART.  260.]  DISCOUNT.  241 

2.  What  is  the  present  worth  of  $250.38,  payable  in 
8  months,  when  money  is  worth  6  per  cent,  per  annum  ? 
What  is  the  discount  ? 

Operation. 

1.04)250.38(240.75  The  amount   of  $1   for  the 

208  given  time  and  rate,  is  $1.04. 

^23  (Art.  247.)     Dividing  the  given 

4  I  Q  sum  by  this  amount,  the  quotient 

$240.75,  is   the  present  worth. 

™*  And    $250.38—240.75=89.63, 

___  the  discount. 

C    $240.75    the    present 
520  Ans.     <          worth ; 

(    $9.63  the  discount. 

3.  What  is  the  present  worth  of  $475,  payable  in    1 
year,  when  money  is  worth  7  per  cent,  per  annum? 

4.  What  is  the  present  worth  of'  $175,  payable  in  2 
years,  when  money  is  worth  7  per  cent,  per  annum  ? 

5.  What  is  the  present  worth  of  $1000,  payable  in  4 
months,  when  the  rate  of  interest  is  6  per  cent.  ? 

6.  What  is  the  discount  on  $750,  due  6  months  hence, 
when  interest  is  5  per  cent,  per  annum  ? 

7.  A  man  sold  a  farm  for  $  1800,  payable  in  15  months : 
what  is  the  present  worth  of  the  debt,  allowing  the  rate 
to  be  6  per  cent.  ? 

8.  I  have  a  note  of  $1150.33,  payable  in  9  months: 
what  is  its  present  worth  at  7  per  cent,  interest  per  an- 
num? 

9.  A  merchant  sold  goods  amounting  to  $840.75,  pay- 
able in  6  months  :  how  much  discount  should  he  make 
for  cash  down,  when  money  is  worth  7  per  cent.  ? 

10.  What  is  the  discount  on  a  draft  of  $2500,  payable 
in  3  months,  at  4£  per  cent,  per  annum  ? 

1 1.  What  is  the  present  worth  of  $5000,  payable  in  2 
months,  at  6  per  cent,  per  annum  ? 

12.  VVhat  is  the  difference  between    the  discount  on 
$500  for  1  year,  and  the  interest  of  $500  for   1  year,  at 
6  per  cent.  ? 


242  DISCOUNT.  [SECT.  IX 


BANK    DISCOUNT. 

261.  It  is  customary  for  Banks  in  discounting  a  no.a 
or  draft,  to  deduct  in  advance  the  legal  interest  on  the  given 
sum  from  the  time  it  is  discounted  to  the  time  when  it 
becomes  due. 

Bank  discount,  therefore,  is  the  same  as  simple  interest 
paid  in  advance.  Thus,  the  bank  discount  on  a  note  of 
$106,  payable  in  1  year  at  6  per  cent.,  is  $6.36,  while  the 
true  discount  is  but  $6.  (Art.  260.) 

OBS.  1.  The  difference  between  bank  discount  and  true  discount,  is 
the  interest  of  the  true  discount  for  the  given  time.  On  small  sums 
for  a  short  period  this  difference  is  trifling,  but  when  the  sum  is  large, 
and  the  time  for  which  it  is  discounted  is  long,  the  difference  is  con- 
siderable. 

2.  Taking  legal  interest  in  advance,  according  to  the  general  rule  of 
law,  is  usury.  An  exception  is  generally  allowed,  however,  in  favor 
of  notes,  drafts,  &c.,  which  are  payable  in  less  than  a  year. 

The  Safety  Fund  Banks  of  the  State  of  New  York,  though  the 
legal  rate  of  interest  is  7  per  cent.,  are  not  allowed  by  their  charters 
to  take  over  6  per  cent,  discount  in  advance  on  notes  and  drafts 
which  mature  within  63  days  from  the  time  they  are  discounted.* 

262*  According  to  custom,  a  note  or  draft  is  not  pre- 
sented for  collection  until  three  days  after  the  time  speci- 
fied for  its  payment.  These  three  days  are  called  days  of 
grace.  It  is  customary  to  charge  interest  for  them. 
Banks,  therefore,  always  calculate  the  interest  for  three 
days  more  than  the  time  stated  in  the  note. 

13.  What  is  the  bank  discount  on  a  note  of  8500,  pay- 
able in  1  year,  at  6  per  cent.  ?  What  is  the  present 
worth  ? 


QUEST. — 261.  How  do  banks  usually  reckon  discount?     What  th«r 
is  bank  discount  ?     Obs.  What  is  the  difference  between  bank  discos 
and  true  discount?     Is  this  difference  worth  noticing  ?     How  is 
interest  in  advance  generally  regarded  in  law  ?     What  exceptica 
this  rule  is  allowed  ?     262.  When  is  it  customary  to  present  r<of-.s  » 
drafts  for  collection  ?     What  are  these  3  days  called  ?     Is  it  c»i« 
to  charge  interest  for  the  days  of  grace  ? 

*  Revised  Statutes  of  New  York,  Vol.  I.  p.  741. 


ARTS.  261, 262.1  DISCOUNT.  243 

Operation. 

The  interest  of  $500  for  1  year  is  $30. 

The         "          "  "  3  days'  grace,  is      0.25 

Therefore  the  discount  is  $30.25 

And  the  present  worth  is  $500 — $30.25=$469.75. 

Note. — Interest  should  be  reckoned  on  the  three  days  grace  in  each 
of  the  following  examples,  except  the  last  two. 

14.  What  is  the  bank  discount  on  a  draft  of  $250, 
payable  in  4  months,  at  7  per  cent.  ? 

15.  What  is  the  bank  discount  on  a  draft  of  $375, 
payable  in  30  days,  at  6  per  cent.  ? 

16.  What  is  the  bank  discount  on  a  note  of  $1000. 
payable  in  60  days,  at  5  per  cent.  ? 

17.  What  is  the  present  worth  of  $1 160,  payable  in  90 
days,  discounted  at  a  bank  at  6  per  cent.  ? 

18.  What  is  the  present  worth  of  $750.36,  payable  in 
f>  months,  at  4£  per  cent.  ? 

19.  What  is  the  bank  discount  of  $  1 825.60,  payable 
in  4  months  and  15  days,  at  6  per  cent? 

20.  What  is  the  present  worth  of  a  draft  of  $1292, 
payable  in  60  days,  at  7  per  cent,  discount? 

21.  What  is  the  present  worth  of  a  draft  of  $5000, 
payable  in  15  days,  at  6  per  cent,  discount? 

22.  What  is  the  present  worth  of  a  draft  of  $15000, 
payable  in  3  days,  at  6  per  cent,  discount? 

23.  What  is  the  present  worth  of  $1326,  payable  in 
10  months,  at  5%  per  cent,  discount? 

24.  What  is  the  bank  discount,  at  7  per  cent.,  on  a 
note  of  $836.81,  payable  in  90  days? 

25.  What  is  the  bank  discount,  at  8  per  cent.,  on  a 
draft  of  $1261.38,  payable  in  60  days? 

26.  What  is  the  bank  discount,  at  6£  per  cent.,  on  a 
draft  of  $10000,  payable  in  30  days? 

27.  What  is  the  difference  between  the  true  discount 
and  bank  discount  on  $1000,  payable  in  5  years,  at  6  per 
cent  ? 

28.  What  is  the  difference  between  the  true  discount 
and  bank  discount  on  $100000,  payable  in  1  year,   at  7 
per  cent.  ? 


244  INSURANCE.  [SECT. 


INSURANCE. 

263.  INSURANCE  is  security  against  loss  or  damage  of 
property  by  fire,  storms  at  sea,  and  other  casualties.     This 
security  is  usually  effected  by  contract  with  Insurance 
Companies,  who,  for  a  stipulated  sum,  agree  to  restore  to 
the  owners  the  amount  insured  on  their  houses,  ships,  and 
other  property,  if  destroyed  or  injured  during  the  specified 
time  of  insurance. 

264.  The  written  instrument  or  contract  is  called  the 
Policy. 

The  sum  paid  for  insurance  is  called  the  Premium. 

The  premium  paid  is  a  certain  per  cent,  on  the  amount 
of  property  insured  for  1  year,  or  during  a  voyage  at  sea, 
or  other  specified  time  of  risk.  Hence, 

265*  To  compute  Insurance  for  1  year,  or  the  speci 
fied  time. 

Multiply  the  sum  insured  by  the  given  rate  per  cent.,  as  in 
interest.  (Art.  237.) 

OBS.  1.  Insurance  on  ships  and  other  property  at  sea  is  sometimes 
effected  by  contract  with  individuals.  It  is  then  called  out-door  in- 
surance. 

2.  The  insurers,  whether  an  incorporated  company  or  individuals! 
are  often  termed  Underwriters. 

Ex.  1.  How  much  premium  must  a  mechanic  pay  an 
nually  for  the  insurance  of  his  shop  and  tools  worth  $350 
at  !-£  per  cent.  ? 

Solution.— $350x.015=$5.25.  Ans. 

2.  What  amount  of  premium  must  be  paid  annually 
for  insuring  a  house  worth  $875,  at  -f  per  cent.  ? 

3.  Shipped  a  box  of  books  valued  at  $1000,  from  New 


QUEST. — 263.  What  is  Insurance?  264.  What  is  meant  by  the 
policy?  The  premium?  265.  How  is  insurance  computed?  Obs. 
When  insurance  is  effected  with  individuals,  what  is  it  called  ?  What 
are  tlxe  insurers  sometimes  called  ? 


ARTS.  263-265.]  INSURANCE,  245 

York  to  New  Orleans,  and  paid  !•}•  per  cent,  insurance . 
tvhat  was  the  amount  of  premium  ? 

4.  A  powder  mill  worth  $925,  was  insured  at  15^  per 
cent. :  what  was  the  annual  amount  of  premium  ? 

5.  A  merchant  shipped  a  lot  of  goods  worth  $1560, 
from  Boston  to  Natchez,  and  paid  1-f-  per  cent,  insurance  ^ 
what  amount  of  premium  did  he  pay  ? 

6.  A  gentleman  obtained  a  policy  of  insurance  on  his 
house  and  furniture  to  the  amount  of  $2500,  at  3-J-  per 
cent,  per  annum :  what  premium  did  he  pay  a  year  1 

7.  A  man  owning  a  sixteenth  of  a  whale  ship,  which 
cost  him  $2750,  got  it  insured,at  7-J-  per  cent,  for  the  voy- 
age :  how  much  did  he  pay  ? 

8.  A  man  owning  a  schooner  worth  $3800,  obtained 
insurance  upon  it,  at  5^  per  cent,  for  the  season :  what 
amount  of  premium  did  he  pay  ? 

9.  A  crockery  merchant  having  a  stock  of  goods  valued 
at  $7500,  paid  2  per  cent,  for  insurance  :  how  much  pre- 
mium did  he  pay  a  year  1 

10.  A  merchant  shipped  83765  worth  of  flour,  from 
Cincinnati  to  New  York,  and  paid  1£  per  cent,  insurance : 
how  much  premium  did  he  pay  ? 

11.  What  is  the  "i^mai  premium  for  insuring  a  store 
worth  $7350,  at  f  per  cent.  ? 

12.  An  importer  effected  insurance  on  a  cargo  of  tea 
worth  $65000,  from  Canton  to  Philadelphia,  at  3  per  cent.  • 
how  much  did  his  insurance  cost  him? 

13.  A  manufacturer  obtained  insurance  to  the  amount 
of  $76500  on  his  stock  and  buildings,  at  -f-  per  cent. :  how 
much  premium  did  he  pay  annually  ? 

14.  A  policy  was  obtained  on  a  cargo  of  goods  valued 
at  $95600,  shipped  from  Liverpool  to  New  York,  at  2-J 
per  cent. :  what  was  the  amount  of  premium  ? 

15.  The  owners  of  the  whale  ship  George  Washing- 
ton obtained  a  policy  of  $58000  on  the  ship  and  cargo,  at 
7\  per  cent,  for  the  voyage :  what  was  the  amount  of 
premium  ? 

16.  A  gentleman  paid  $60  annually  for  insurance  on 
his  house  and  furniture,  which  was  2  per  cent,  on  its  value 
what  amount  of  property  was  covered  by  the  policy? 


246  INSURANCE.  [SECT.  IX , 

Note. — Tliis  example  is  similar  to  those  of  Problem  III,  in  interest 
(Art.  254.) 

Solution. — Since  the  rate  of  insurance  is  2  per  cent.  01 
.02,  it  is  plain  that  $60  is  -rjhr  of  the  amount  insured. 
Now  if  $60  is  -rihr,  ifa  is  half  as  much,  or  $30 ;  and 
•H-4  is  $30X100.  or  $3000.  Or  thus:  60^.02=3000 
Ans.  $3000. 

PROOF.— $3000x.02=$60,  which  was  the  annual  pre- 
mium paid. 

17.  If  I  pay  $250  premium  on  silks,  from  Havre  to 
New  York,  at  !-£-  per  cent.,  what  amount  of  property 
does  my  policy  cover  ? 

18.  A  merchant  paid  $1200  premium,  at  2-^  per  cent, 
on  a  ship  and  cargo  from   London  to  Baltimore,  which 
was  lost  on  the  voyage :  what  amount  should  he  recover 
from  the  Insurar.ee  Company  ? 

19.  If  a  man  pays  $60  premium  annually  for  the  in- 
surance of  his  house,  which  is  worth  $3000,  what  rate 
per  cent,  does  he  pay  ? 

Note. — This  example  is  similar  to  those  of  Problem  II,  in  interest 
(Art.  253.) 

Solution.— $60-*-$3000=.02.  Ans.  2  per  cent. 

PROOF. — $3000x.02=$60,  which  is  the  premium  paid. 

20.  A  merchant  paid  $40  premium  for  insuring  $5000 
on  his  stock  :  what  rate  per  cent,  did  he  pay? 

21.  If  a  man  pays  $75  for  insuring  $15000,  what  rate 
per  cent,  does  he  pay  1 

22.  If  the  owner  pays  $2800  for  insuring  a  ship  worth 
$40000,  what  rate  per  cent,  does  he  pay  ? 

23.  A   blacksmith  owns   a  shop   worth    $720:  what 
amount  must  he  get  insured  annually,  at  10  per  cent,  so 
that  in  case  of  loss,  both  the  value  of  the  shop  and  the. 
premium  may  be  repaid  ? 

Analysis. — Since  the  rate  of  insurance  is  10  per  cent., 
on  a  policy  of  $100,  the  owner  would  actually  receive 
but  $90  ;  for  he  pays  $  1 0  for  insurance.  The  question 
then  resolves  itself  into  this:  $720  is  -rVg  of  what  sum? 


ART.  266.]  PROFIT  AND  LOSS.  247 

If  720  is-flfo  -rh-  is  720+90=8,  ai  1  ^0  is  8x100= 
800.  Ans.  $800. 

PROOF. — $800X-10=$80,  the  premium  he  would  pay, 
and  $800 — $80=$720,  which  is  the  value  of  his  shop. 

24.  If  I  send  an  adventure  to  China  worth  $6250, 
what  amount  of  insurance,  at  8  per  cent.,  must  I  obtain, 
that  in  case  of  a  total  wreck  I  may  sustain  no  loss  by  the 
operation  1 

25.  What  amount  of  insurance  must  be  effected  on 
$1 1250,  at  5  per  cent.,  in  order  to  cover  both  the  premium 
and  property  insured  ? 

PROFIT  AND  LOSS. 

266*  PROFIT  and  Loss  in  commerce,  signify  tb3  sum 
gained  or  lost  in  ordinary  business  transactions.  They 
are  reckoned  at  a  certain  per  cent,  on  the  purchase  price, 
or  sum  paid  for  the  articles  under  consideration. 


MENTAL    EXERCISES. 

1.  A  merchant  bought  a  barrel  of  flour  for  $6,  and  sold 
it  at  a  profit  of  10  per  cent. :  how  much  did  he  sell  it  for  ? 

Suggestion. — Since  he  made  10  per  cent,  profit,  if  we 
add  10  per  cent,  to  the  purchase  price,  it  will  give  the 
selling  price.  Now  10  per  cent,  of  $6  is  60  cents,  (Art 
225,)  which  added  to  $6,  make  $6.60. 

Ans.  He  sold  it  for  $6.60. 

2.  A  grocer  bought  a  box  of  oranges  for  $5,  and  sold  it, 
at  12  per  cent,  profit :  how  much  did  he  receive  for  his 
oranges  ? 

3.  A  farmer  bought  a  ton  of  hay  for  $9,  and  sold  it 


QUEST— 266.  What  is  meant  by  profit  and  loss  ?     How  are  they 
reckoned  ? 


248  PROFIT  AND   LOSS.  [SECT.  IX» 

for  10  per  cent,  more  than  he  gave :  how  much  did  he 
sell  it  for? 

4.  Bought  a  sleigh  for  $12,  and  sold  it  at  a  loss  of  8 
per  cent. :  how  much  did  I  receive  for  the  sleigh  ? 

Solution. — 8  per  cent,  of  $12,  is  96  cents;  and  $12 — 
96  cents  leaves  $11.04.  Ans. 

5.  Bought  a  box  of  honey  for  $5,  and  having  lost  a 
portion  of  it,  sold  the  remainder,  at  1 1  per  cent,  loss :  how 
much  did  I  receive  for  it  ? 

6.  A  shop-keeper  bought  a  piece  of  calico  for  $7,  and 
sold  it,  at  12  per  cent,  profit :  how  much  did  he  sell  it  for  ? 

7.  A  lad  bought  a  sheep  for  $3,  and  on  his  way  home 
was  offered  15  per  cent,  for  his  bargain:  how  much  was 
he  offered  for  his  sheep  ? 

8.  A  farmer  bought  a  colt  for  $20,  and  offered  to  sell 
it  for  5  per  cent,  less  than  he  gave:  how  much  did  he 
ask  for  it? 

9.  A  gentleman  bought  a  horse  for  $100  ;  after  using 
it  awhile,  he  sold  it,  at  7  per  cent,  loss :  how  much  did  he 
get  for  his  horse  ? 

10.  A  man  bought  a  building  lot  for  $150,  and  in  con 
sequence  of  the  rise  of  property,  sold  it  for  10  per  cent, 
advance :  how  much  did  he  get  for  it? 

11.  A  hack-man  bought  a  carriage  for  $200,  and  after 
using  it  for  one  season,  sold  it  for  15  per  cent,  less  than 
he  gave  for  it :  how  much  did  he  sell  it  for  ? 

12.  A  man  bought  a  house  for  $800,  and  sold  it  the 
next  day  for   10  per  cent,   advance :  how  much  did  he 
sell  it  for  ? 


EXERCISES    FOR     THE- SLATE. 
CASE   I. 

i.  A  merchant  bought  a  quantity  of  grain  for  $75,  and 
sold  it  for  8  per  cent,  profit  :  how  much  did  he  gain  bj 
the  bargain  ? 

Solution.— $75X.08=$6.00.  (Art.  225.)     Hence, 


ART.  267.]  PROFIT  AND  LOSS.  249 

267*  To  find  the  amount  of  profit  or  loss,  when  the 
purchase  price  arid  rate  per  cent,  are  given. 

Multiply  the.  purchase  price  by  the  given  per  cent,  as  in 
percentage ;  and  the  product  will  be  the  amount  gained  or 
lost  by  the  transaction.  (Art.  225.) 

2.  A  man  bought  a  sleigh  for  $60,  and  afterwards  sold 
it  for  10  per  cent,  less  than  cost :  how  much  did  he  lose  ? 

3.  A  grocer  bought  a  cask  of  oil  for  $96.50,  and  re- 
tailed it,  at  a  profit  of  6  per  cent. :  how  much  did  he 
make  on  his  oil  ? 

4.  A  pedlar  bought  a  lot  of  goods  for  $215,  and  retail- 
ed them,  at  20  per  cent,  advance :  how  much  was  his 
profit? 

5.  A  merchant  bought  a  cargo  of  coal  for  $450,  which 
he  afterwards  sold  for  12-£  per  cent,  less  than   cost:  what 
was  the  amount  of  his  loss'? 

6.  A  manufacturer  purchased  $1000  worth  of  wool, 
and  after  making  it  up,  sold  the  cloth  for  25  per  cent, 
more  than  the  cost  of  the  materials :  how  much  did  he 
receive  for  his  labor  1 

CASE    II. 

7.  A  man  bought  a  span  of  horses  for  $350,  and  wished 
to  dispose  of  them  for  12  per  cent,  profit:  how  much 
must  he  sell  them  for? 

Operation.  Reasoning  as  before,  he 

$350  purchase  price,   must  sell  them  for  the  jmr- 

.12  per  cent,  profit,  chase  price,    together    with 

$42.00  gained.  12  per  cent,  of  that  price. 

^w«~«toQQsplliTi0-iirirp        Having  found  12  per  cent. 

[mg  price'       of  $350,    (Art.    225,)    add 

ft  to  the  cost,  and  the  sum  $392,  is  manifestly  the  soling 

vrice. 

8.  A  stage  proprietor  bought  a  coach  for  $480 ;  find- 

QUEST. — 267.  How  is  the  amount  of  profit  or  loss  found,  when  th« 
tost  and  rate  per  cent,  are  given  ? 


PROFIT  AND  LOSS.  [SECT.  IX, 

ing  it  damaged,  he  was  willing  to  sell  it,  at  5  per  cent 
loss  :  at  wha*  price  would  he  sell  it  ? 

Operation.  Having   found   the   sum 

$480  purchase  price,  lost,  (Art.  225,)  subtract  it 

•05  per  cent.  loss,  from  the  cost,  and  the   Te- 

$24.00  sum  lost  mainder    is    obviously  *h*» 

sellin      rice.  sellinS  Price'     flence> 


268.  To  find  ho~v  ary  article  must  be  sold,  in  orde,» 
to  gain  or  lose  a  given  rate  per  cent. 

First  find  the  amount  of  profit  or  loss  on  the  purcha**. 
price  at  the  given  rate,  as  in  the  last  Case  ;  then  the  amou*v 
thus  found  added  to,  or  subtracted  from  the  'purchase  price 
as  the  case  may  be,  will  give  the  selling  price  required. 

9.  A  merchant  bought  a  firkin  of  butter  for  $22.75- 
how  much  must  he  sell  it  for  in  order  to  gain  15  per  cent 
by  his  bargain  ? 

10.  Bought  a  chest  of  tea  for  $37.50:  for  how  mucb 
must  I  sell  it,  in  order  to  make  18  per  cent,  by  the  opera 
tion? 

11.  Bought  a  quantity  of  produce  for  $89.33,  which  3 
propose  to  sell,  at  20  per  cent,  loss  :  how  much  must  } 
receive  for  it  ? 

12.  A  drover  bought  a  flock  of  sheep  for  $275,  am 
taking  them  to  market,  sold  them,  at  25  per  cent,  ad 
vance  :  how  much  did  he  sell  them  for  ? 

13.  A  merchant  had  a  quantity  of  groceries  on  hand 
which  cost  him  $367.13;  for  the  sake  of  closing  up  hh 
business  he  sold  them,  at  15  per  cent,  less  than  cost  :  how 
much  did  he  get  for  them  ? 

14.  A  man  bought  a  farm  for  $875,  and  was  offered 
33  per  cent,  advance  for  his  bargain  :  how  much  was  he 
offered? 

15.  A  merchant  bought  a  cargo  of  cotton  for  $30000; 


QUEST. — 268.  What  is  the  method  of  finding  how  an  article  must  b* 
sold,  in  order  to  p-iin  or  lose  a  given  per  cent.  ? 


ARTS.  268,  269. J      PROFIT  AND  LOSS  251 

the  price  declining,  he  sold  it  at  2£  per  cent,   less  than 
cost :  for  how  much  did  he  sell  it  ? 

CASE    III. 

16.  A  man  bought  a  cow  for  $25,  which  he  afterwards 
<old  for  $29  :  what  per  cent,  profit  did  he  make  ? 

Analysis. — Subtracting  the  cost  from  the  selling  price, 
shows  that  he  gained  $4.  Now  4  dollars  are  -fa  of  25 
dollars ;  hence,  he  gained  -fa  of  his  outlay,^  the  purchase 
price  of  the  cow.  And  -fa  reduced  to  a  decimal,  is  16 
hundredths,  which  is  the  same  as  16  per  cent.  (Arts.  197 
223.  Obs.  3.) 

Or,  we  may  reason  thus :  If  25  dollars  (outlay)  gain  4 
dollars,  1  dollar  (outlay)  will  gain  fa  of  4  dollars.  Now 
$4-5-25  is  equal  to  16  hundredths  of  a  dollar.  But  16 
hundredths  is  the  same  as  16  per  cent.  Hence, 

269.  To  find  the  rate  per  cent,  of  profit  or  loss,  when 
the  cost  and  selling  prices  are  given. 

First  fi'nd  the  amount  gained  or  lost,  as  the  case  may  be, 
by  subtraction ;  then  make  the  gain  or  loss  the  numerator 
and  the  purchase  price  the  denominator  of  a  common  fraction, 
which  being  reduced  to  a  decimal,  will  give  the  per  cent,  re- 
quired. (Art.  197.) 

Or,  simply  annex  ciphers  to  the  profit  or  loss,  and  divide 
it  by  the  cost ;  the  quotient  will  be  the  per  cent. 

OBS.  1.  As  per  cent,  signifies  hundredths,  we  have  seen  that  the 
first  tico  decimal  figures  which  occupy  the  place  of  hundredths,  are 
properly  the  per  cent. ;  the  other  decimals  are  parts  of  1  per  cent. 
After  obtaining  two  decimal  figures,  there  is  sometimes  an  advantage 
in  placing  the  remainder  over  the  divisor,  and  annexing  it  to  the  de- 
cimals thus  obtained.  (Art.  223.  Obs.  3.) 

2.  It  should  be  remembered  that  the  percentage  which  is  gained  or 
»ast,  is  always  calculated  on  the  purchase  price,  or  the  sum  paid  for 


QUEST. — 2G9.  How  is  the  rate  per  cent,  of  profit  or  loss  found,  when 
the  cost  and  selling  price  are  given  ?  Obs.  What  figures  properly  sig- 
nify the  per  cent.  ?  Why  ?  What  do  the  other  decimal  figures  on  the 
right  of  »undredths  denote  ?  On  what  is  the  per  cent,  gained  or  loe> 


252  PROFIT   AND    LOSS.  [SECT.  IX 

the  article,  and  not  on  the  selling  price,  or  sum  received,  as  it  is  often 
supposed. 

17.  A  merchant  bought  a  piece  of  cloth  for  $2.75  pei 
yard,  and  sold  it  for  $3.25 :  what  per  cent,  did  he  gain  'I 

Solution. — Since  he  gained  50  cents  on  a  yard,  his  gain 
was  -gfe  of  the  cost.     And  •£&=.  1  S-ft. 

Ans.  18-fV  per  cent. 

18.  A  boy  purchased  a  book  for  20  cents,  and  sold  jt 
for  30  cents :  what  per  cent,  did  he  make  ? 

19.  A  merchant  bought  a  box  of  sugar,  at  6  cents  a 
pound,  and  sold  it  for  7-J-  cents  a  pound  :  what  per  cent, 
was  his  profit  ? 

20.  A  grocer  bought  eggs  at  9  cents,  and  sold  them  for 
12  cents  per  dozen  :  what  per  cent,  was  his  profit? 

21.  A  man  bought  a  hat  for  $4.50,  and  sold  it  for  $6 : 
what  per  cent,  did  he  gain  ? 

22.  A  jockey  bought  a  horse  for  $73,  and  sold  him 
for  $68 :  what  per  cent,  did  he  lose? 

23.  A  merchant  bought  a  quantity  of  goods  for  $155.63 
and  sold  them  for  $148.28  :  what  per  cent,  did  he  lose? 

24.  A  gentleman  bought  a  house  for  $3500,  and  sold 
it  for  $150  more  than  he  gave  :  what  per  cent,  was  his 
profit? 

25.  A  speculator  laid  out  $7500  in  land,  and  afterwards 
sold  it  for  $10000  :  what  per  cent,  did  he  make  ? 

26.  A  drover  bought  a  herd  of  cattle  for  $1175,  and 
sold  them  for  $1365:  what  per  cent,  did  he  gain;  and 
how  much  did  he  make  by  the  operation  ? 

27.  A  merchant  bought  $10000  worth  of  wool,  and 
sold  it  for  $12362:  what  per  cent. ;  and  how  much  was 
his  profit? 

CASE   IV. 

28.  A  jockey  sold  a  horse  for  $250,  which  was  25  per 
cent,  more  than  it  cost  him :  how  much  did  he  pay  for  tho 
horse  ? 

Analysis. — It  will  be  observed  that  the  selling  price 
($250)  is  equal  to  the  cost  and  the  amount  gained  added 


ART.  270.  j  PROFIT  AND  LOSS.  253 

together.  Now  considering  the  cost  a  unit  or  1,  the  gain 
which  is  a  certain  per  cent,  of  the  cost,  (Art.  266,)  is  T^-, 
consequently  l+^^-Hnh  (Art.  127,)  will  denote  the 
sum  of  the  cost  and  the  gain.  The  question  therefore 
resolves  itself  into  this  :  250  is  +*%  of  what  number  ?  If 
250  is  -HHh  -rb-  is  2 ;  and  iffr  is  100  times  2,  or  200. 

Or;  we  may  simply  divide  250  by  the  fraction  -ftf-ft. 
(Art.  141.)  The  quotient  200  is  the  cost  required. 

PROOF.— $200x.25=$50  ;  and  $200+$50=$250,  the 
selling  price  ? 

29.  A  merchant  sold  a  quantity  of  goods  for  $180, 
which  was  1 0  per  cent,  less  than  cost :  how  much  did  the 
goods  cost  him  ? 

Analysis. — It  will  be  observed  that  the  selling  price 
($180)  is  equal  to  the  cost  diminished  by  the  sum  lost. 
Now  reasoning  as  in  the  last  example,  1 — rVu^iW  wiM 
denote  the  cost  diminished  by  the  loss.  The  question  now 
is  this :  180  is  -^  of  what  number  ?  If  180  is  T2^-,  rb 
is  2,  and  -HH}  is  200.  Or  thus:  S180^13D%-=$200.  Ans. 

PROOF.— $200x.  10=820,  and  $200— $20-$180,  the 
selling  price.  Hence, 

27O.  To  find  the  cost  when  the  selling  price  and  the 
per  cent,  gained  or  lost  are  given. 

Make  the  given  per  cent,  added  to  or  subtracted  from  100, 
as  the  case  may  be.  the  numerator,  and  100  the.  denominator 
of  a  common  fraction;  then  divide  the  selling  price  by  this 
fraction  ;  aiid  the  quotient  will  be  the  cost  required. 

OBS.  1.  It  is  not  unfrequently  supposed  that  if  we  find  the  per- 
centage on  the  selling  price  at  the  given  rate,  and  add  the  percentage 
thus  found  to,  or  subtract  it  from  the  selling  price,  as  the  case  may  be, 
the  sum  or  remainder  will  be  the  cost.  This  is  a  mistake,  and  leads 


QUEST.— 270.  How  is  the  cost  found,  when  the  selling  price  and  the 
rate  per  cent,  gained  or  lost,  are  given  ?     Obs.  What  mistake  is  iome 
made  in  finding  the  cost  ?    How  may  it  be  avoided  ? 


s 

254  PROFIT  AND  LOSS.  [SECT.    IX. 

to  serious  errors  in  the  /esult.  It  will  easily  be  avoided  by  remem- 
bering, that  the  basis  on  which  profit  and  loss  are  calculated,  i* 
always  the  purchase  price,  or  sum  paid  for  the  articles  under  con- 
sideration. (Art.  2b'9.  Obs.  2.) 

30.  A  grocer  sold  a  hogshead  of  molasses  for  $24,  and 
gained  20  per  cent,  on  the  cost :  what  was  the  cost  of  the 
molasses  ? 

31.  A  merchant  sold  a  piece  of  broadcloth  for  $85, 
which  was  10  per  cent,  less  than  the  cost:  what  was  the 
cost  of  it  ? 

32.  A  butcher  sold  a  yoke  of  oxen  for  $125,  and  there- 
by made  15  per  cent.  :  how  much  did  they  cost  him  1 

33.  A  bookseller  sold  a  lot  of  books  for  $200,  which 
was  12  per  cent,  more  than  the  cost:  what  was  the  cost? 

34.  A  wholesale  druggist  sold  a  quantity  of  medicines 
for  $560,  and  made  50^  per  cent,   profit  on  them  :  what 
was  the  cost  of  them  ? 

35.  A  merchant    sold     a    cargo  of  rice  for  $1500, 
which  was  12£  per  cent,  less  than  cost:  what  was  the 
cost? 

EXAMPLES   FOR.   PRACTICE. 

1.  A  merchant  bought  25  boxes  of  raisins  for  $45  :  af 
what  price  per  box  must  he  retail  them  to  gain  10  per 
cent,  by  his  bargain  ? 

Suggestion. — He  must  sell  the  whole  for  10  per  cent 
more  than  the  cost.  Hence,  if  we  add  10  per  cent,  to 
the  cost,  and  divide  the  sum  by  the  number  of  boxes,  it 
will  give  the  retail  price  per  box.  (Art.  217.) 

2.  A  shopkeeper  bought  a  piece  of  cotton  containing 
40  yards,  at  6   cents  a  yard,  and  sold  it  for  7  cents  a 
yard :  what  per  cent,  profit  did  he  gain  ;  and  how  much 
did  he  make  by  the  bargain  ? 

3.  A  merchant  bought  60  yards  of  domestic  flannel  at 
25  cents   per  yard,  and  sold   it  at  30  cents  per  yard : 
what  per  cent,  was  his  profit ;  and  how  much  did  he  clear 
by  the  operation  ? 

4.  A  bookseller  bought  100  Arithmetics  at  Slfrcentf 


ART.  270.]  PROFIT  AND  LOSS.  255 

apiece,  and  retailed  them  at  37£  cents  apiece  :  what  per 
cent. ;  and  how  much  did  he  make  by  the  operation. 

5.  A  drover  bought  175  sheep  for  $350',  and  sold  them 
?o  as  to  gain  15  per  cent. :  how  much  did  he  sell  them 
for  per  head  1 

6.  A  baker  paid   $2500  for  480  barrels  of  flour,  and 
finding  it  damaged,  sold  it  at  a  loss  of  8  per  cent.  :  how 
much  did  he  sell  it  for  per  barrel  1 

7.  A  merchant  bought  10  pieces  of  broadcloth,  each 
piece  containg  30  yards,  for  $1400,  and  retailed  the  whole 
at  a  profit  of  20  per  cent. :  at  what  price  did  he  sell  it  per 
yard? 

8.  A  grocer  bought  500  Ibs.  of  butter  for  $75,  and  sold 
it  at  a  loss  of  7  per  cent. :   how  much  did  he  get  per 
pound  1 

9.  A  merchant  bought  12  hogsheads  of  molasses  at  25 
cents  per  gallon :  how  must  he  sell  it  by  the  gallon  in 
order   to   gain  20  per  cent. ;  and   how  much   was   his 
profit  ? 

10.  A  farmer  raises  750  bushels  of  wheat  at  an  ex 
perise  of  $675  :  how  must  he  sell  it  per  bushel,  in  order 
to  make  18  per  cent.  ? 

1 1.  A  provision  merchant  bought  1500  barrels  of  pork 
at  $10.25  per  barrel,  and  sold  it  at  a  loss  of  9  per  cent.  : 
how  much  did  he  lose  ;  and  what  did  he  get  per  barrel  ? 

12.  An  inn-keeper  bought   150  bushels  of  oats,  at  25 
cents  a  bushel,  and  retailed  them  at  the  rate  of  12£  cents 
a  peck :  what  per  cent. ;  and  how  much  did  he  make  on 
the  oats  ? 

13.  A  miller  bought  500  bushels  of  wheat,  at  75  cents 
per  bushel :  how  much  must  he  sell  the  whole  for  in  order 
to  gain  20  per  cent.  ? 

14.  A  grocer  bought  1630  pounds  of  tea,  at  62-^-  cents 
per  pound,  and  sold  it  at  10  percent,  loss  :  how  much  did 
he  sell  it  at  per  pound  ? 

15.  A  merchant  bought  a  bale  of  calico  prints  contain- 
ing 750  yards  and  paid  $75 :  how  must  he  retail  it  per 
yard,  in  order  to  gain  20  per  cent. ;  and  how  much  would 
he  make  on  a  yard  ? 

16.  A  bookseller  purchased    1000  geographies,  at  84 


256  DUTIES.  [SECT.  IX. 

cents  apiece :  how  must  he  retail  them  to  gain  20  pei 
cent.  ? 

17.  A  milliner  bought  1200  yards  of  ribbon,  at  30 
cents  per  yard :  how  must  she  sell  it  per  yard  to  gain  50 
per  cent.  ? 

1 8  A  grocer  bought  5000  Ibs.  of  sugar  for  $350,  and 
retailed  it,  at  6  cents  per  pound  :  what  per  cent,  loss  did 
he  sustain  ? 

19.  A  man  purchased  goods  amounting  to    $1635: 
what  per  cent,  profit  must  he  gain,  in  order  to  make  $350  ? 

20.  A   speculator   bought"    10000   acres    of  land   for 
$12500,  and  afterwards  sold*  it,  at  25  per  cent,  loss:  for 
how  much  per  acre  did  he  sell  it ;  and  how  much  did  he 
lose  by  the  operation  ? 

DUTIES. 

271.  DUTIES,  in  commerce,  signify  a  sum  of 
required  by  Government  to  be  paid  on  imported  goods. 

Duties  are  of  two  kinds,  specific  and  ad  valorem.  A 
specific  duty  is  a  certain  sum  imposed  on  a  ton,  hundred 
weight,  hogshead,  gallon,  square  yard,  foot,  &c.  without 
regard  to  the  value  of  the  article. 

Ad  valorem  duties  are  those  which  are  imposed  on 
goods,  at  a  certain  per  cent,  on  their  value  or  purchase 
price. 

Note. — The  term  ad  valorem  is  a  Latin  phrase,  signifying  according 
to,  or  upon  the  value. 

272.  Before  specific  duties  are  imposed,  it  is  custo- 
mary to  make  certain  deductions  called  tare,  draft,  or 
tret,  leakage,  &c, 

Tare,  in   commerce,  is    an  allowance  of   a  certain 


QUEST. — 271.  What  are  duties  in  commerce  ?  Of  how  many  kind* 
are  they  ?  What  are  specific  duties  \  Ad  valorem  duties  ?  Note. 
What  is  the  meaning  of  the  term  ad  valorem  ?  272.  What  deductions 
are  made  before  specific  duties  are  imposed  ?  What  is  tare  ?  Draft 
or  tret  ?  Leakage  ? 


271-273.]  DUTIES.  257 

number  of  pounds  made  for  the  box,  cask,  &c.,  which 
contains  the  article  under  consideration. 

Draft  or  Tret  is  an  allowance  of  a  certain  per  cent, 
(usually  4  per  cent.)  on  the  weight  of  goods  for  waste,  or 
refuse  matter. 

Leakage  is  an  allowance  of  a  certain  per  cent,  (usually 
2  per  cent.)  for  the  waste  of  liquors  contained  in  casks,  &c. 

OBS.  1 .  All  duties,  both  specific  and  ad  valorem,  are  regulated  by 
the  Government,  and  have  been  different  at  different  times  and  in 
different  countries. 

2.  The  allowance  or  deductions  for  draft,  tare,  leakage,  &c.,  are 
also  different  on  different  articles,  and  are  regulated  by  law. 

3.  In  buying  and  selling  groceries  in  large  quantities,  allowances 
are  sometimes  made  for  draft,  tare,  leakage,  &c.,  similar  to  those  in 
reckoning  duties. 

CASE    I. 

Ex.  1.  What  is  the  specific  duty  on  10  pipes  of  wine, 
at  15  cents  per  gallon,  reckoning  the  leakage  at  2  per 
cent.  1 

Suggestion. — First  deduct  the  leakage.  In  1  pipe  there 
are  2  hogsheads  or  126  gallons ;  in  10  pipes  there  are  10 
times  126,  or  1260  gallons.  But  2  per  cent,  of  1260 
gallons,  is  1260x.02-25.20  gallons;  (Art.  225 ;)  and 
25.2  gallons  subtracted  from  1260  gallons  leaves  1234.8 
for  the  number  of  net  gallons.  Now  if  the  duty  on  1  gal- 
lon is  15  cents,  on  1234.8  gallons  it  is  1234.8x.l5= 
$185.22,  the  duty  required.  Hence, 

273.  To  fin'  the  specific  duty  on  any  given  merchan- 
dise. 

First  deduct  the  legal  draft,  tare,  leakage,  &c.  from  the 
given  quantity  of  goods  ;  then  multiply  the  remainder  by  the 
given  duty  per  gallon,  pound,  yard,  <$-«.,  and  the  product 
will  be  the  duty  required, 

QUEST. — Obs.  How  are  duties  regulated  ?  Ar<J  the  allowance*  fat 
draft,  tare,  &c.  the  same  for  all  articles  I  Are  allowances  ever  made 
in  buying  and  selling  groceries  for  draft,  &o.  1  273.  How*  are  specific 
duties  calculated  ? 


258  DUTIES.  [SECT.  IX 

2.  What  is  the  specific  duty,  at  2  cents  per  pound,  on 
12  boxes  of  sugar,  weighing  900  Ibs.  apiece,  allowing  20 
pounds  per  box  for  draft  ? 

A       <  The  draft  is  240  pounds. 

?>   \  And  2  per  ct.  on  10560  Ibs.  is  $211.20. 

3.  At  3  cents  a  pound,  what  is  the  duty  on  25  casks  of 
nails,  each  weighing   125  Ibs.   allowing  8  pounds  on  a 
cask  for  tare  ? 

4.  At  5  cents  a  pound,  what  is  the  specific  duty  on  75 
boxes  of  raisins,  weighing  60   Ibs.   apiece,  allowing  6 
pounds  a  box  for  draft  ? 

5.  At  4  cents  per  pound,  what  is  the  specific  duty  on 
110  chests  of  cinnamon,  each  weighing  230  Ibs.  allowing 
16  Ibs.  per  chest  for  draft? 

6.  At  15  cents  a  pound,  what  is  the  specific  duty  on 
300  bags  of  indigo,  each  weighing  200  Ibs.,  allowing  4 
per  cent,  for  tret  ? 

CASE    II. 

7.  What  is  the  ad  valorem  duty,  at  15  per  cent,  on  an 
invoice  of  calico  prints,  which  cost  $150  in   Liverpool? 

Suggestion. — When  duties  are  imposed  upon  the  actual 
cost  of  merchandise,  there  are  of  course  no  deductions  to 
be  made  ;  consequently  we  have  only  to  find  15  per  cent, 
of  $150,  the  amount  of  the  given  invoice,  or  cost  of  the 
goods,  and  it  will  be  the  duty  required. 

Solution.— $150x-15-$22.50.  Ans.  Hence, 

274.  To  find  the  ad  valorem  duty  on  any  given  mer- 
chandise. 

Multiply  the  given  invoice  by  the  given  or  legal  per  cent, 
and  the  product  will  be  the  duty  required.  (Art.  225.) 

OBS.  1.  An  invoice  is  a  written  statement  of  merchandise,  with  the 
value  or  prices  of  the  articles  annexed. 

QUEST. — 274.  How  are  ad  valorem  duties  calculated  ?  Obs.  What 
i>  an  invoice?  What  does  the  law  require  respecting  the  invoiced 
•aaported  goods  ? 


ART.  274.]  DUTIES.  259 

2.  The  law  requires  that  the  invoice  shall  be  verified  by  the  owner, 
or  one  of  the  owners  of  the  goods,  wares,  or  merchandise,  certifying 
that  the  invoice  annexed  contains  a  true  and  faithful  account  of  tht 
actual  costs  thereof,  and  of  all  charges  thereon,  and  no  othei  differ- 
ent discount,  bounty,  or  drawback,  but  such  as  has  been  actually  al- 
lowed on  the  same;  which  oath  shall  be  administered  by  a  consul,  or 
commercial  agent  of  the  United  States,  or  by  some  public  officer  duly 
authorized  to  administer  oaths  in  the  country  where  the  goods  were 
purchased,  and  the  same  shall  be  duly  certified  by  the  said  consul,  &c. 
Fraud  on  the  part  of  the  owners,  or  the  consul,  &c.  who  administers 
the  oath,  is  visited  with  a  heavy  penalty. — Laws  of  Hie  United  States. 


8.  What  is  the  ad  valorem  duty,  at  30  per  cent,  on  a 
box  of  books  invoiced  at  $250  ? 

9.  What  is  the  ad  valorem  duty,  at  20  per  cent.,  on  a 
quantity  of  Java  coffee,  which  cost  $356.12? 

10.  What  is  the  amount  of  ad  valorem  duty,  at  25  per 
cent,    on  a  quantity  of   Turkey  carpeting,    which  cost 
$526.61. 

11.  What  is  the  duty  on  a  quantity  of  bombazines,  in- 
voiced at  $310,  at  30  per  cent.  ? 

12.  What  is  the  duty  on  a  quantity  of  beeswax,  the  in 
voice  of  which  is  $460.25,  at  15  per  cent,  ? 

13.  At  25  per  cent,  what  is  the  duty  on  an  invoice  oi 
bleached  linens,  amounting  to  $745.85. 

14.  At  20  per  cent,  what  is  the  duty  on  an  invoice  of 
jewelry,  amounting  to  $4250  ? 

15.  W'hat  is  the  duty  on  a  bale  of  goods,  invoiced  at 
$2500,  at  40  per  cent.  ? 

16.  What  is  the  duty  on  an  invoice  of  silks,  amounting 
to  $5650,  at  30  per  cent.  ? 

17.  What  is  the  duty  on  a  quantity  of  cutlery,  invoiced 
at  $4560,  at  33  per  cent  ? 

18.  What  is  the  duty  on  an  invoice  of  broadcloths, 
which  amounts  to  $8280,  at  35  per  cent.  ? 

19.  What  is  the  duty  on  an  invoice  of  wines,  amount- 
ing to  $10265,  at  35  per  cent? 

20.  What  is  the  duty  on  a  quantity  of  cotton  fabrics, 
invoiced  at  S13637.50,  at  33  per  cent? 

21.  What  is  the  duty  on  a  quantity  of  ready-made 
clothing,  amounting  to  $5638.25,  at  50  per  cent  ? 


\ 

260  ASSESSMENT  [SECT.  IX 


ASSESSMENT  OF  TAXES. 

275.  A  TAX  is  a  sum  imposed  or  levied  on  indi 
riduals  for  the  support  or  benefit  of  the  Government,  a 
corporation,  parish,  district,  &c.  Taxes  levied  by  the 
Government,  are  assessed  either  on  the  person  or  property 
of  the  citizens.  When  assessed  on  the  person,  they  are 
called  poll  taxes,  and  are  usually  a  specific  sum.  Those 
assessed  on  the  property  are  usually  apportioned  at  a  cer- 
tain per  cent,  on  the  amount  of  real  estate  and  personal 
property  of  each  citizen  or  taxable  individual. 

OBS.  Property  is  divided  into  two  kinds,  viz :  real  estate,  and  per- 
sonal property.  The  former  denotes  possessions  that  are  fixed ;  as 
houses,  lands,  &c.  The  latter  comprehends  all  other  property;  as 
money,  stocks,  notes,  mortgages,  ships,  furniture,  carriages,  cattle, 
tools,  &c. 

276*  When  a  tax  of  any  given  amount  is  to  be  as- 
sessed, the  first  thing  to  be  done  is  to  obtain  an  inventory 
01'  the  amount  of  taxable  property,  both  personal  and 
real,  in  the  State,  County,  Corporation,  or  District,  by 
which  the  tax  is  to  be  paid ;  also  the  amount  of  property 
of  every  citizen  who  is  to  be  taxed,  together  with  the 
number  of  Polls. 

OBS.  1.  By  the  number  of  polls  is  meant  the  number  of  taxable 
individuals,  which  usually  includes  every  native  or  naturalized  free- 
man over  the  age  of  21,  and  under  70  years.  In  some  States  it  also 
includes  the  young  men  over  the  age  of  eighteen  years,  who  are  sub- 
ject to  military  duty. 

2.  When  any  part  or  the  whole  of  a  tax  is  assessed  upon  the  polls, 
each  citizen  is  taxed  a  specific  sum,  without  regard  to  the  amount  of 
property  he  possesses. 

Ex.  1.  A  certain  town  is  taxed  $325.  The  town  con- 
tains 200  polls,  which  are  assessed  25  cents  apiece  ;  and 


QUEST. — 275.  What  are  taxes  t  Upon  what  are  they  assessed  ? 
When  assessed  upon  the  person,  what  are  they  called  ?  When  assessed 
upon  the  property,  how  are  they  apportioned  ?  Obs.  How  is  pro^  j"y 
ui\kled?  W  Ii.it  does  real  estate  denote  ?  What  is  personal  property  ? 
$76.  When  a  tax  is  to  be  assensed,  what  is  the  first  step  ?  Oft*.  Wl.at 
is  meant  by  the  number  of  polls  ? 


ARTS.  275-278.]  OF  TAXES.  261 

the  whole  amount  of  property  both  real  and  personal,  is 
valued  at  $13750.  How  much  is  the  tax  on  a  dollar; 
that  is,  what  per  cent,  is  the  tax,  and  how  much  is  a 
man's  tax  who  pays  for  1  poll,  and  whose  property  is 
valued  at  $850  ? 

Suggestion. — The  tax  on  the  polls  is  200x.25=$50. 
And  $50  subtracted  from  $325  leaves  $275,  which  is  to 
be  assessed  equally  on  the  amount  of  property  possessed 
by  the  citizens  of  the  town.  The  next  step  is  to  find  how 
much  must  be  paid  on  a  dollar.  Now  if  $13750  pay 
$275,  $1  must  pay  ia}50  part  of  $275.  And  $275-*- 
$13750=$.02,  the  tax  on  $1,  which  is  2  per  cent.  Fi- 
nally, at  2  per  cent.,  or  2  cents  on  $1,  the  tax  on  $850, 
the  amount  of  the  man's  property,  is  $850x.02=$  17.00. 
And  $17+.25  (the  poll)=$  17.25,  the  man's  tax.  Hence, 

27  7  •  To  assess  a  State,  County,  or  ether  tax. 

1.  First  find  the  amount  of  tax  on  all  the  polls,  if  any, 
at  the  given  rate,  and  subtract  this  sum  from  the  whole  Lax 
to  be  assessed.     Then  dividing  the  remainder  by  the  whole 
amount  of  taxable  property  in  the  Stale,  County,  fyc.,  the 
quotient  will  be  the  per  cent,  or  tax  on  1  dollar. 

II.  Multiply  the  amount  of  each  man's  property  by  the 
per  cent,  or  tax  on  one  dollar,  and  the  product  will  be  the  tax 
vn  his  property. 

III.  Add  each  marts  poll  tax  to  the  tax  he  pays  on  his 
property,  and  the  amount  will  be  his  whole  tax. 

2  7  8 .  PROOF. —  When  a  tax  bill  is  made  out,  add  together 
the  taxes  of  all  the  individuals  in  the  town,  district,  <fyc., 
and  if  the  amount  is  equal  to  the  whole  tax  assessed,  the  work 
is  right. 

2.  A  certain  parish  is  taxed  $237.50.      The  whole 
property  of  the  parish  is  valued  at  $8000  ;  and  there  are 
75  polls,  which  are  assessed  50  cents  apiece.     What  per 
cent,  is  the  tax ;  and  how  much  is  a  man's  tax  who  pays 
for  3  polls,  and  whose  property  is  valued  at  $500  ? 

QUEST. — 277.  How  arc  taxes  assessed  ?    378.  When  a  tax  bill  is 
maile  out,  htnv  is  its  correctness  proved  ? 


262  ASSESSMENT.  [SECT. 

Operation. 

First  multiply  .50  cents,  the  tax  on  1  }  oil, 
By  75  the  number  of  polls. 

$37.5(J  amount  on  polls. 

Then  $237.50— $37.50=$200,  the  sum  to  be  assessed 
en  the  property. 

$80eOO)$200.000(.0255  the  per  cent,  or  tax  on  $1. 
16000 


40000 
40000 

And  $500x025=$  12.50,  the  tax  on  the  man's  property. 
.50x3=   1.50,  tax  for  polls. 
Ans.  $14.00,  his  whole  tax. 

3.  What  amount  of  tax  does  a  man  living  in  the  same 
parish  pay,  whose  property  is  valued  at  $450,  and  pays 
for  2  polls  ? 

4.  A  tax  of  $750  is  assessed  on  a  district  to  build  a 
new  school-house ;  the  property  of  the  district  is  valued 
at  $15000.      What  is  the  tax  on  a  dollar;  and  what  is 
a  man's  tax  whose  property  is  $1150? 

5.  What  is  B's  tax  for  erecting  the  same  school-house, 
whose  property  is  $1530? 

6.  A  tax  of  $14752.50  is  levied  on  a  certain  County, 
whose  property  is  valued  at  $562875,  and  which  has  a 
list  of  5825  polls,  which  are  assessed  at  60  cents  apiece. 
What  per  cent,  is  the  tax ;  and  what  is  the  amount  of  C's 
,ax,  who  pays  for  4  polls,  and  has  property  valued  at 
$5000  ? 

7.  What  is  D's  tax,  who  living  in  the  same  County, 
pay-s  for  2  polls,  and  is  worth  $3500  ? 

8.  What  is  G's  tax,  who  pays  for  5  polls,  and  is  worth 
$15300? 

279,  In  making  out  a  tax  bill  for  a  whole  town,  dis- 
trict, &c.,  assessors,  having  found  the  tax  on  $1,  usually 
make  a  table,  showing  the  amount  of  tax  on  any  number  w 


279.1 


OF    TAXES. 


263 


dollars  from  1  to  $10  ;  then  on   10,  20,  30,  &c.  to  $100; 
then  on  100,  200.  &c.  to  $1000. 

9.  A  tax  of  $3506.25  was  levied  on  a  corporation  cooi 
posed  of  12  individuals,  whose  property  was  valued  at 
$175000,  and  who  were  assessed  for  25  polls  at  25  cents 
apiece.  What  was  the  tax  on  a  dollar  ? 

Ans.  2  cents  on  a  dollar. 

Note. — Having  found  the  tax  on  $1,  we  will  make  a  table  to  aid 
us  in  making  out  the  tax  bill  of  the  corporation.  Since  the  tax  on 
$1  is  $.02,  it  is  obvious  that  multiplying  $.02  by  2  will  be  the  tax  on 
$2 ;  multiplying  it  by  3,  will  be  the  tax  on  S3,  &c. 

TABLE. 


$1  pays  $.02 

$10  pays  $.20 

SI  00  pays  $2.00 

2  «   .04 

20   "    .40 

200  «    4.00 

3  "   .06 

30   "    .60 

300  "    6.00 

4  "   .08 

40   «    .80 

400  "   8.00 

5  "   .10 

50   «   1.00 

500  «   10.00 

6  «   .12 

60   «   1.20 

600  "   12.00 

7  "   .14 

70   «   1.40 

700  "   14.00 

8  "   .16 

80   «   1.60 

800  «   16.00 

9  "   .18 

90   "   1.80 

900  «   18.00 

10  "   .20 

100   «   2.00 

4000  "   20.00 

10.  In  the  above  assessment,  what  was  A's  tax,  whose 
property  was  valued  at  $1256,  and  who  pays  for  3  polls? 

Operation. 

$1256  is  composed  of  1000-f- 
200-J-50-T-6.  Now,  if  we  add 
the  taxes  paid  on  each  of  these 
sums  together,  the  amount  will 
be  the  tax  paid  on  $1256. 

A's  tax,  therefore,  was  $25.87. 

11.  What  was  B's  tax,  who  paid  for  4  polls,  and  had 
property  to  the  amount  of  $1461 1 

1 2.  C  paid  for  1  poll,  and  the  valuation  of  his  property 
was  $5863.     What  was  the  amount  of  his  tax  ? 


$1000  pays  $20.00 

200    «          4.00 

50    «  1.00 

6    "  .12 

3  polls    "  .75 

Amount,  $25.87. 


QUEST. — 279.  When  a  tax  bill  is  to  be  made  out  for  a  whole  town, 
distriet,  &c.,  what  course  do  assessors  usually  take  ? 


264  PROPERTIES  [SECT. 

1 3.  D  paid  for  1  poll,  and  the  valuation  of  his  property 
was  $7961.     What  was  his  tax? 

1 4.  E  paid  for  2  polls,  and  his  property  was  valued  at 
$  1 4236.     What  was  his  tax  ? 

15.  F  paid  for  2  polls,  and  his  real  estate  was  valued 
at  $21000  ;  his  personal  property  at  $4500.     What  was 
his  tax  ? 

16.  G's  property  was  valued  at  $20250,  and  he  paid 
for  1  poll.     What  was  his  tax  ? 

17.  H  paid  for  2  polls,  and  the  valuation  of  his  estate 
was  $  1 5360.     What  was  his  tax  ? 

18.  J's  property  was  valued  at  $33000,  and  he  paid  foi 
4  polls.     What  was  his  tax  ? 

19.  K  paid  for  1  poll,  and  his  property  was  valued  at 
$  1 50 1 3.     What  was  his  tax  ? 

20.  L  paid  for  3  polls,  and  his  property  was  valued  at 
$4500.     What  was  his  tax  ? 

21.  M  paid  for  1  poll,  and  the  valuation  of  his  property 
was  $30600.     What  was  his  tax  ? 


SECTION    X. 
PROPERTIES    OF    NUMBERS.* 

DEFINITIONS. 

ART.  28O.  The  progress  as  well  as  the  pleasure  of 
the  pupil  in  the  study  of  Arithmetic,  depends  very  much 
upon  the  accuracy  of  his  knowledge  of  the  terms,  which 
are  employed  in  mathematical  reasoning.  Hence,  par- 
ticular care  has  been  taken  to  define  all  the  most  impor- 
tant terms,  as  they  have  been  introduced,  and  it  is  of  the 
utmost  importance  for  the  pupil  to  understand  their  true 
import. 


QtTEST. — 280.  Upon  what  does  the  progress  and  pleasure  of  the  stu 
dent  in  Arithmetic  very  much  depend  ? 

*  Barlow  en  the  Theory  of  Numbers 


ART.  280.]  OF  NUMBER*  265 

DBF.  1 .  Numbers  are  divided  into  two  classes,  abstract, 
and  concrete. 

When  they  are  applied  to  particular  objects,  as  peaches, 
pounds,  yards,  &c.,  they  are  called  concrete. 

When  they  are  not  applied  to  any  particular  object, 
they  are  called  abstract.  (Art.  45.  Obs.  1.)  Thus,  when 
it  is  said  that  two  and  three  are  five,  the  two,  three,  and 
{ice  denote  abstract  numbers. 

2.  An  integer  signifies  a  whole,  number.  (Art.  105.) 

3.  Whole  numbers  or  integers  are  divided  into  prime 
and  composite  numbers. 

4.  A  prime  number  is  one  which   cannot  be  produced 
by   multiplying   any   two   or   more    numbers   together. 
Thus,  1,  2,  3,  5,  7,  11,  13,  17,  19,  23,  29,  31,  &c.,  are 
prime  numbers. 

Osa.  1.  A  prime  number  is  exactly  divisible  only  by  itself  and  a 
unit. 

2.  One  number  is  said  to  be  prime  to  another,  when  a  unit  is  the 
only  number  by  which  both  can  be  divided  without  a  remainder. 

The  number  of  prime  numbers  is  unlimited.  The  first  twelve  are 
given  above.  The  pupil  can  easily  point  out  others. 

5.  A  composite  number  is  one  which  may  be  produced 
by  multiplying  two  or  more  numbers  together.  (Art.  55. 
Obs.  1.)     Thus,  4,  6,  8,  9,  10,  12,  14,   15,  16,  &c.  are 
composite  numbers. 

6.  An  even  number  is  one  which  can  be  divided  by  2 
without  a  remainder  ;  as,  4,  6.  8,  10. 

7.  An  odd  number  is  one  which  cannot  be  divided  by 
2  without  a  remainder;  as,  1,  3,  5,  7,  9,  15. 

OBS.  All  even  numbers  except  2,  are  composite  numbers ;  an  odd 
aumber  is  sometimes  a  c&mposile,  and  sometimes  a  prime  number. 

QUEST. — Into  how  many  classes  are  numbers  divided  ?  What  is  an 
abstract  number  ?  A  concrete  number  1  What  is  an  integer  ?  Into 
how  many  classes  are  whole  numbers  divided  ?  What  is  a  prime  num- 
ber I  Obs.  Are  prime  numbers  divisible  by  other  numbers  ?  When  is 
sne  number  said  to  be  prime  to  another  I  How  many  prime  numbers 
are  there  ?  What  is  a  composite  number  ?  What  is  an  even  number  ? 
An  odd  number  I  Obs.  Are  even  numbere  prime  or  composite  1  What 
ia  true  of  odd  numbp.-s  in  this  respect  ? 


266  PROPERTIES.  [SECT. 

8.  One  number  is  a  measure  of  another,  when  the  for 
mer  is  contained  in  the  latter  any  number  of  times  witliou; 
a  remainder.  (Art.  93.  Obs.  1.) 

9.  One  number  is  a  multipk  of  another,  when  the  for 
mer  can  be  divided  by  the  latter  without  a   remainder 
(Art.  98.) 

10.  The  aliquot  parts  of  a  number  are  the  parts  by 
which  it  can  be  measured,  or  into  which  it  may  be  divided 
Thus,  3  and  7  are  the  aliquot  parts  of  21. 

11.  The  reciprocal  of  a  number  is  the  quotient  arising 
from  dividing  a  unit  by  that  number.     Thus,  the  recip- 
rocal of  2  is  l-*-2,  or  •£ ;  the  reciprocal  of  3  is  l-*-3,  or  -^. 


PROPERTIES     OP    THE     SUMS,     DIFFERENCES,     PRO- 
DUCTS, &c.,   OF  NUMBERS. 

28  1,  By  properties  of  numbers,  is  meant  those  qual- 
ities or  elements  of  numbers  which  are  inseparable  from 
them. 

1.  The  sum  of  any  two  or  more  even  numbers,  is  an 
even  number. 

2.  The  difference  of  any  two  even  numbers,  is  an  even 
number. 

3.  The  sum  or  difference  of  two  odd  numbers,  is  even , 
but  the  sum  of  three  odd  numbers,  is  odd. 

4.  The  sum  of  any  even  number  of  odd  numbers,  is 
even  ;  but  the  sum  of  any  odd  number  of  odd  numbers, 
is  odd. 

5.  The  sum,  or  difference,  of  an  even  and  an  odd  num- 
ber^  is  an  odd  number. 

6.  The  product  of  an  even  and  an  odd  number,  or  of 
two  even  numbers,  is  even. 

7.  If  an  even  number  be  divisible  by  an  odd  number, 
the  quotient  is  an  even  number. 

8.  The  product  of  any  number  of  factors,  is  even^  if 
any  one  of  them  be  even. 

9.  An  odd  number  cannot  be  divided  by  an  even  num 
her  without  a  remainder. 

QUEST. — When  is  one  number  a  measure  of  another  ?  When  i» 
one  number  a  multiple  of  another  ?  What  are  aliquot  parts  ?  What 
is  the  reciprocal  of  a  number?  281.  What  is  meant  by  properties  oi 
numben  ? 


L/ 

A  Tt  rpc 


281, 282.]        OP  NUMBERS.  267 

10.  The  product  of  any  two  or  more  odd  numbers,  is  an 
udd  number. 

11.  If  an  odd  number  divides  an  even  number,  it  will 
also  divide  the  half  of  it. 

12.  If  an  even  number  is  divisible  by  an  odd  number, 
it  will  also  be  divisible  by  double  that  number. 

13.  The   product  of  any  two  numbers   is  the   same, 
whichever  of  the  two  numbers  is  the  multiplier.  (Art.  47.) 

14.  The  least  divisor  of  every  number,  is  a  prime  num- 
ber. 

OBS.  Hence,  in  obtaining  the  least  common  multiple,  the  smallest 
number  which  will  divide  any  two  or  more  of  the  given  numbers,  is 
always  a  prime  number,  and  consequently  we  divide  by  a  prime  num- 
ber. (Art.  102.) 

15.  Any  number  expressed  by  the  decimal  notation, 
divided  by  9,  will  leave  the  same  remainder  as  the  sum  of 
its  figures  or  digits  divided   by  9.     The  same  property 
belongs  to  the  number  3,  and  to  no  other  number.     Thus, 
if  236  is  divided  by  9,  the  remainder  is  2  ;  so,  if  the  sum 
of  its  digits,  2-j-3-[-6=^ll,  is  divided  by  9,  the  remainder 
is  also  2. 

Note. — Upon  this  property  of  the  number  9,  is  based  a  convenient 
method  of  proving  multiplication  and  division. 

PROOF  OF   MULTIPLICATION   BY  CASTING  OUT  THE 

NINES. 

282.  First,  cast  the  9s  out  of  the  multiplicand  and  mul- 
tiplier ;  multiply  their  remainders  together,  and  cast  the  9s 
out  of  their  product,  and  set  down  the  excess  ;  then  cast  the  9s 
out  of  the  ansioer  obtained,  and  if  this  excess  be  the  same  as 
that  obtained  from  the  multiplier  and  the  multiplicand,  tJie 
work  may  be  considered  right. 

Note. — To  cast  out  the  9s  from  a  number,  begin  at  the  left  hand, 
add  the  digits  together,  and  as  soon  as  the  sum  is  9  or  over,  drop  the 


QUEST. — What  is  the  least  divisor  of  every  number?  Obs.  In  ob- 
taining the  least  common  multiple  of  two  or  more  numbers,  by  what 
tind  of  a  number  do  we  divide  ?  282.  How  is  multiplication  proved  by 
»asting  out  the  9s  ? 


268  PROPERTIES  [SECT. 


9,  and  add  the  remainder  to  the  next  digit,  and  so  on.  For  exam- 
ple, to  cast  the  9s  out  of  4026357,  we  proceed  thus :  4  and  6  are  10-, 
drop  the  9  and  add  the  I  to  the  next  figure.  1  and  2  are  3,  and  6 
are  9  ;  drop  the  9  as  above.  3  and  5  are  8  and  7  are  15;  drop  the 
9,  and  we  have  6  remainder. 

Multiply  565  by  356. 

Operation.  Proof 

565     The  excess  of  9s  in  the  multiplicand  is  7 
356       "         «          9s      "      multiplier  is       5. 
3390     7x5=35  ;  and  the  excess  of  9s  is  8 

2825 
1695 

Prod.  201140.    The  excess  of  9s  in  the>  Ans.  is  also  8. 

PROOF  OF  DIVISION   BY  CASTING  OUT  THE  NINES. 

S83.  First  cast  the  9.5  out  of  the  divisor  and  quotient^ 
and  multiply  the  remainders  together  ;  to  the  product  add  the. 
remainder,  if  any,  after  division  ]  cast  the  9s  out  of  this  sum. 
and  set  down  the  excess ;  finally  cast  theQs  out  of  the  dividend, 
and  if  the  excess  is  the  same  as  that  obtained  from  the  divisor 
and  quotient,  the  work  may  be  considered  right. 

AXIOMS. 

284«  In  mathematics,  there  are  certain  propositions 
whose  truth  is  so  evident  at  sight,  that  no  process  of  rea- 
soning- can  make  it  plainer.  These  propositions  are 
called  axioms. 

•An  axiom*  therefore,  is  a  self-evident  proposition. 

1.  Quantities  which  are  equal  to  the  same  quantity,  are 
equal  to  each  other. 

2.  If  the  same  or  equal  quantities  are  added  to  equai 
quantities,  the  sums  will  be  equal. 

3.  If  the  same  or  equal  quantities  are  subtracted  from 
equals,  the  remainders  will  be  equal. 

4.  If  the  same  or  equal  quantities  are  added  to  unequal^, 
the  sums  will  be  unequal. 

QUEST. — 283.  How  is  division  proved  by  casting  out  the  9s? 


.  283-286.]          OF  NUMBERS.  369 

5.  If  the  same  or  equal  quantities  are  subtracted  from 
U'/iequals,  the  remainders  will  be  unequal. 

6.  If  equal  quantities  are  multiplied  by  the  same  or 
equal  quantities,  the  products  will  be  equal. 

7.  If  equal  quantities  are  divided  by  the  same  or  equal 
quantities,  the  quotients  will  be  equal. 

8.  If  the  same  quantity  is  both  added  to  and  subtracted 
from  another,  the  value  of  the  latter  will  not  be  altered 

9.  If  a  quantity  is  both  multiplied  and  divided  by  the 
same  or  an  equal  quantity,  its  value  will  not  be  altered. 

10.  The  whole  of  a  quantity  is  greater  than  a  part. 

11.  The  'whole  of  a  quantity  is  equal  to  the  sum  of  all 
its  parts. 

OBS.  The  term  q uantity  signifies  any  thing  which  can  be  multiplied, 
divided,  or  measured.  Thus,  numbers,  yards,  busJids,  weight,  time, 
&c.,  are  called  quantities. 

285.  The  following  principles  will  at  once  be  recog- 
nized by  the  pupil  as  deductions  from  the  four  Fundamen- 
tal Rules  of  Arithmetic,  viz  :  Addition,  Subtraction,  Mul- 
tiplication, and  Division. 

286.  When  the  sum  of  two  numbers  and  one  of  the 
numbers  are  given,  to  find  the  other  number. 

From  the  given  sum  subtract  the  given  number,  and  the 
remainder  will  be  the.  other  number. 

Ex.  1.  The  sum  of  two  numbers  is  25,  and  one  01 
them  is  10  ;  what  is  the  other  number? 

Solution. — 25 — 10=15,  the  other  number.  (Art.  40.) 
PROOF. — 15-1-10=25,  the  given  sum.  (Art.  284.  Ax.  1 1.) 

2.  A  and  B  together  own  36  cows,  9  of  which  belong 
to  A :  how  many  does  B  own  ? 

3.  Two  farmers  bought  300  acres  of  land  together,  and 
one  of 'them  took   115  acres:  how  many  acres  did  the 
other  have  ? 


QUEST. — 284.  What  is  an  axiom  ?  What  is  the  first  axiom  ?  The 
second?  Third?  Fourth?  Fifth?  Sixth?  Seventh?  Eighth?  Ninth! 
Tenth  ?  Eleventh  ?  Obs.  What  is  meant  by  quantity  ?  286.  When  the 
mm  of  two  numbers  and  cne  of  them  «r«  riven,  how  ia  the  other  found  f 


270  PROPERTIES  [SECT. 

287*  When  the   difference  and  the   greater  of  two 
numbers  are  given,  to  find  the  less. 

Subtract  the  difference  from  the  greater ,  and  the  remainder 
icill  be  the  less  number. 

4.  The  greater  of  two  numbers  is  37,  and  the  difference 
between  them  is  10:  what  is  the  less  number? 

Solution. — 37 — 10=27,  the  less  number.  (Art.  40.) 

PROOF. — 274-10=37,  the  greater  number.     (Art.  39. 
Obs.) 

5.  A  had  48  dollars  in  his  pocket,  which  was  12  dollars 
more  than  B  had  :  how  many  dollars  had  B  ? 

6.  D  had  450  sheep,  which  was  63  more  than  E  had : 
kow  many  had  E  ? 

289.  When  the  difference  and  the  less  of  two  num- 
bers are  given,  to  find  the  greater. 

Add  the  difference  and  less  number  together,  and  the  sum 
icill  be  the  greater  number.  (Art.  39.) 

7.  The  difference  between  two  numbers  is  5,  and  the 
less  number  is  15:  what  is  the  greater  number? 

Solution. — 154-5=20,  the  greater  number. 

PROOF. — 20 — 15=5,  the  given  difference.  (Art.  40.) 

8.  A  is  16  years  old,  and  B  is  8  years  older:  how  old 
isB? 

9.  The  number  of  male  inhabitants  in  a  certain  town, 
is  935  ;  and  the  number  of  females  exceeds  the  number 
of  males  by  115:  how  many  females  does  the  town  con- 
tain? 


QUEST. — 287.  When  the  difference  and  the  greater  of  two  numbeia 
are  given,  how  is  the  less  found  ?  289.  When  the  difference  and  *fcfl 
less  of  two  numbers  are  given,  how  is  the  greater  found  ? 


AJB/S.  287-291.]        OF  NUMBERS,  27 1 

29O.  When  the  sum  and  difference  of  two  numbers 

ij-e  given,  to  find  the  two  numbers. 

From  the  sum  subtract  the  difference,  and  half  the  remainder 
till  be  the  smaller  number. 

To  the  smaller  number  thus  found,  add  the  given  difference, 
and  the  sum  will  be  the  larger  number. 

10.  The  sum  of  two  numbers  is  35,  and  their  difference 
is  1 1 :  what  are  the  numbers  ? 

Solution.— 35—  1 1=24  ;  and  £  of  24=12,  the  smaller 
number.  And  12+11=23,  the  greater  number. 

PROOF. — 23+12=35,  the  given  sum.  (Art.  284, 
Ax.  11.) 

1 1.  The  sum  of  the  ages  of  2  boys  is  25  years,  and  the 
difference  between  them  is  5  years :  what  are  their  ages  ? 

12.  A  man  bought  a  chest  of  tea  and  a  hogshead  of 
molasses  for  $63  ;  the  tea  cost  $9  more  than  the  molasses : 
what  was  the  price  of  each  ? 


When  the  product  of  two  numbers  and  one  o 
the  numbers  are  given,  to  find  the  oilier  number. 

Divide  the  given  product  by  the  given  number,  and  the 
quotient  will  be  the  number  required.  (Art.  74.) 

.    13.  The  product  of  two  numbers  is  84,  and  one  of  the 
numbers  is  7  :  what  is  the  other  number  ? 

Solution. — 84-7-7=12,  the  required  number.  (Art.  72.) 
PROOF. — 12x7=84,  the  given  product.  (Art.  54.) 

14.  The  product  of  A  and  B's  ages  is  120  years,  and 
A's  age  is  12  years :  how  old  is  B  ? 

15.  A  certain   field  contains    160  square  rods,  and  the 
length  of  the  field  is  20  rods:  what  is  its  breadth  ? 


QUEST.— 290.  When  the  sum  and  difference  of  two  numbers  nre 
given,  how  are  the  numbers  found?  291.  When  the  product  of  two 
numbers  and  one  of  them  are  given,  how  is  the  other  found  ? 


272  PROPERTIES  [SECT.  jfl 

Note.  —  The  area  of  a  field  is  found  by  multiplying  its  length  and 
breadth  together.  (Art.  163.)  Hence  the  area  of  a  field  may  be  con- 
•idered  as  a  product 

292.  When  the  divisor  and  quotient  are  given  to  find 
the  dividend. 

Multiply  the  given  divisor  and  quotient  together,  and  tht 
•oduct  will  be  the  dividend.  (Art.  73.) 

16.  If  a  certain  divisor  is  9,  and  the  quotient  is  12 
what  is  the  dividend  ? 

Solution.  —  12x9=108,  the  dividend  required. 
PROOF.  —  108-^-9=12,  the  given  quotient.  (Art.  72.) 

17.  A  man  having  1  1  children,  gave  them  $75  apiece  • 
how  many  dollars  did  he  give  them  all  ? 

18.  A  farmer  divided  a  quantity  of  apples  among  90 
ooys,  giving  each  boy  15  apples  :  how  many  did  he  give 
them  all  ? 


When  the  dividend  and  quotient  are  given,  to 
find  the  divisor. 

Divide  the  given  diviflend  by  the  given  quotient,  and  the. 
quotient  thus  obtained  will  be  the  number  required.  (Art.  73. 
Obs.  2.) 

19.  A  certain  dividend  is  130,  and  the  quotient  is  10: 
what  is  the  divisor  ? 

Solution.  —  130-^-10=13,  the  divisor  required.  (Art.  72.) 
PROOF.—  13x10==-  130,  the  given  dividend.  (Art.  73.) 

20.  A  gentleman  divided  $120  equally  among  a  com- 
pany of  sailors,  giving  them  $10  apiece  :  how  many  sail- 
ors were  there  in  the  company  ? 


QUEST. — 292.  When  the  divisor  and  qxiotient  are  given,  how  is  tha 
dividend  found  ?  293.  When  the  dividend  and  quotient  are  given  hov 
HJ  the  divitor  found  ! 


ARTS.  292-295.]         oy  NUMBERS,  273 

21.  A  farmer  having  600  sheep,   divided  them  into 
flocks  of  75  each :  how  many  flocks  had  he  ? 

294*  When  the  product  of  three  numbers  and  iico  of 
the  numbers  are  given,  to  find  the  other  number. 

Divide  the  given  product  by  the  product  of  the  two  given 
numbers,  and  the  quotient  witt  be  tJie  other  number. 

22.  There  are  three  numbers  whose  product  is  60 ; 
one  of  them  is  3,  and  another  5  :  it  is  required  to  find  the 
other  number? 

Solution. — 5x3=15;  and   60-1-15=4,  the   number  re- 
quired. 

PROOF. — 5x3x4=60,  the  given  product. 

23.  The  product  of  A,  B,  and  C's  ages,  is  210  years  ; 
the  age  of  A  is  5  years,  and  that  of  B  is  6  years :  what 
is  the  age  of  C  ? 

24.  The  product  of  three  boys'  marbles,  is  1728  ;  two 
of  them  have  a  dozen  apiece  :  how  many  has  the  other? 


SECTION    XI. 

ANALYSIS. 

ART.  29  5  •  Business  men  have  a  method  of  solving 
practical  questions,  which  is  frequently  shorter  and  more 
expeditious,  than  that  of  arithmeticians  fresh  from  the 
schools.  If  asked,  by  what  rule  they  perform  them,  their 
reply  is,  "  they  do  them  in  their  head"  or  by  the  "  no 
rule  method" 

Their  method  consists  in  Analysis,  and  may,  with 
propriety,  be  called  the  COMMON  SENSE  RULE. 


QUEST — 294.  When  the  product  of  three  numbers  and  two  of  them 
are  given,  how  is  the  other  found  ?  295.  What  is  said  of  the  method 
by  which  business  men  solve  practical  questions  ?  In  what  does  their 
method  consist  ?  What  may  it  with  propriety  be  called  ? 


274  ANALYSIS.  [SECT. 

The  term  analysis,  in  physical  science,  signifies  the  re> 
tolving  of  a  compound  body  into  its  elements  or  compo* 
nent  parts. 

ANALYSIS,  in  Arithmetic,  signifies  the  resolving  of  num- 
bers into  the  factors  of  which  they  are  composed,  and  the 
tracing  of  the  relations  which  they  bear  to  each  other. 

OBS.  In  the  preceding  sections  the  student  has  become  ac- 
quainted with  the  method  of  analyzing  particular  examples  and  com" 
binations  of  numbers,  and  thence  deducing  general  principles  and 
rides.  But  analysis  may  be  applied  with  advantage  not  only  to  the 
development  of  mathematical  truths,  but  also  to  the  solution  of  a  great 
variety  of  problems  both  in  arithmetic  and  practical  life. 


MENTAL    EXERCISES. 


Ex.  1.  If  8  barrels  of  flour  cost  $40,  how  much  will 
5  barrels  cost  1 

Analysis.  —  1  is  1  eighth  of  8  :  therefore  1  barrel  will 
cost  1  eighth  as  much  as  8  barrels  ;  and  I  eighth  of  $40 
is  $5.  Now  it  is  obvious  that  5  barrels  will  cost  5  times 
as  much  as  1  barrel  ;  and  5  times  $5  are  $25,  the  answer 
required. 

Or,  we  may  reason  thus  ;  5  barrels  are  -f  of  8  barrels  ; 
5  barrels  will  therefore  cost  -f  as  much  as  8  barrels.  Now 
1  eighth  of  $40  is  $5,  and  5  eighths  is  5  times  $5,  which 
is  $25.  Ans. 

2.  If  7  Ibs.  of  tea  cost  42  shillings,  what  will  10  Ibs 
cost? 

3.  If  9  sheep  are  worth  $27,  how  much  are  15  sheep 
worth  ? 

4.  If  10  barrels  of  flour  cost  $60,  what  will  12  barrels 
cost? 

5.  Suppose  30  gallons  of  molasses  cost  $15,  how  many 
dollars  will  7  gallons  cost  ? 


QUEST.  —  295.  a.  What  is  meant  by  analysis  in  physical  science  1 
What  in  arithmetic  ?  Obs.  To  what  may  analysis  be  advantageously 
applied  ? 


.  295.  a.\  ANALYSIS.  275 

6.  If  a  man  earns  54  shilling?  m  6  days,  how  much 
can  he  earn  in  15  days  1 

7.  If  12  men  can  build  48  rods  of  wall  in  a  day>  how 
juany  rods  can  20  men  build  in  the  same  time  1 

8.  A  gentleman  divided  90  shillings  equally  among 
15  beggars:  how  many  shillings  did  7  of  them  receive? 

9.  Suppose  75  pounds  of  butter  last  a  family  of  board- 
ers 25  days,  how  many  pounds  will  supply  them  for  12 
days? 

*10.  If  7  yards  of  cloth  cost  $30,  how  much  will  9  yards 
cost? 

11.  If  10  barrels  of  beef  cost  $72,  how  much  will  8 
barrels  cost  ? 

12.  If  7  acres  of  land  cost  $50,  what  will  12  acres 
cost? 

13.  A  farmer  bought  an  ox  cart,  and  paid  $15  down, 
which  Avas  -fo  of  the  price  of  it :  what  was  the  price  of 
the  cart ;  and  how  much  does  he  owe  for  it  ? 

Analysis. — The  question  to  be  solved  is  simply  this  . 
15  is  -A-  of  what  number  ?  If  15  is  -fV,  -^  is  ±  of  15, 
which  is  5.  Nt>w  if  5  is  1  tenth,  10  tenths  is  10  times 
5,  which  is  50. 

.        ^  $50  is  the  price  of  the  cart,  and 
'•    }  $50— $15-35,  the  sum  unpaid. 

Note. — In  solving  examples  of  this  kind,  the  learner  is  often  per- 
plexed in  finding  the  value  of  -JL-,  &c.  This  difficulty  arises  from 
supposing  that  if  -fc  of  a  certain  number  is  15,  -fo  of  it  must  be  -j-L- 
of  15.  This  mistake  will  be  easily  avoided  by  substituting  in  his 
mind  the  word  parts  for  the  given  denominator. 

Thus,  if  3  parts  cost  $15,  1  part  will  cost  .1  of  $15,  which  is  $5. 
But  this  part  is  a  tenth.  Now  if  1  tenth  cost  $5,  then  10  tenths  will 
cost  1 0  times  as  much. 

14.  A  man  bought  a  yoke  of  oxen,  and  paid  $56  cash 
down,  which  was  •£  of  the  price  of  them  :  what  did  they 
cost  2 

15.  A  merchant  bought  a  quantity  of  wood  and  paid 
$45  in  goods,  which  was  f  of  the  whole  cost :  how  much 
iid  he  pay  for  the  wood  ? 

16.  A  whale  ship  having  been  out  24  months,  the  cap 


276  ANALYSIS.  [SECT. 

tain  found  that  his  crew  had  consumed  -f  of  his  provis- 
ions  :  how  many  months'  provision  had  he  when  he  em- 
barked ;  and  how  much  longer  would  his  provisions  last  ? 

17.  How  many  times  7  in  -f-  of  35? 

Analysis. — |-  of  35  is  7,  and  £  is  4  times  7,  which  is 
28.     Now  7  is  contained  in  28, 4  times.     Ans.  4  times. 

18.  How  many  times  6  in  f  of  45  ? 

19.  How  many  times  10  in  f  of  60? 

20.  How  many  times  12  in  -f-  of  84  ? 

21.  -f  of  42  are  how  many  times  6  ? 

22.  •£  of  40  are  how  many  times  5  ? 

23.  -^  of  80  are  how  many  times  12  ? 

24.  f  of  48  are  how  many  times  4  ? 

25.  •£  of  64  are  how  many  times  7  ? 

26.  T^-  of  100  are  how  many  times  12  ? 

27.  -ft  of  110  are  how  many  times  8  ? 

28.  -f  of  180  are  how  many  times  10  ? 

29.  -fa  of  84  are  how  many  times  9  ?- 

30.  How  many  yards  of  cloth,  at  $7  per  yard,  can  be 
bought  for  i  of  $54  ? 

31.  How  many  barrels  of  flour,  at  $5  per  barrel,  can 
be  bought  for  -f  of  $60  ? 

32.  A  man  had  $64  in  his  pocket,  and  paid  -f-  of  it  for 
10  barrels  of  flour :  how  much  was  that  per  barrel  ? 

33.  40  is  -f-  of  how  many  times  6  ? 

Analysis. — Since  40  is  f,  i  is  •§•  of  40,  or  8  ;  and  f  is 
9  times  8,  or  72.     Now  6  is  contained  in  72,  12  times. 

Ans,  12  times 

34.  56  is  -f  of  how  many  times  7  ? 

35.  81  is  T^T  of  how  many  times  30  ? 

36.  72  is  T8!-  of  how  many  times  9  ? 

37.  96  is  -f-  of  how  many  times  12  ? 

38.  64  is  -fo  of  how  many  times  20  ? 

39.  54  is  -f  of  how  many  times  24  ? 

40.  108  is  -ft-  of  how  many  times  12  ? 


.  296.]  ANALYSIS.  277 

41.  Frank  sold  10  peaches,  which  was  f  ol  all  he  had ; 
He  then   divided  the  remainder  equally  among  5  com- 
panions: how  many  did  they  receive  apiece? 

42.  Lincoln  spent  60  cents  for  a  book,  which  was  if 
of  his  money  ;  the  remainder  he  laid  out  for  oranges,  at 
4  cents  apiece :  how  many  oranges  did  he  buy  ? 

43.  A  man  paid  away  $35,  which  was  f  of  all  he  had  : 
he  then  laid  out  the  rest  in  cloth  at  $2  per  yard :  how 
many  yards  did  he  obtain  ? 

44.  A  farmer  bought  a  quantity  of  goods,  and  paid  $20 
down,  which  was  -f  of  the  bill :  how  many  cords  of  wood, 
at  $3  per  cord,  will  it  take  to  pay  the  balance  ? 

45.  A  man  bought  a  horse  and  paid  $60  in  cash,  whi-;h 
was  f  of  the  price  :  how  many  barrels  of  flour  at  $6  per 
barrel,  will  it  take  to  pay  the  balance? 

46.  ij-  of  27  is  f  of  what  number  ? 

Analysis. — f  of  27  is  9.  And  if  9  is  £  of  a  certain 
number,  -|  of  that  number  is  3  ;  and  f  is  4  times  3,  which 
ss  12,  the  number  required, 

47.  -rV  of  30  is  f  of  what  number  ? 

48.  -f-  of  40  is  -f-  of  what  number  ? 

49.  4-  of  35  is  -rV  of  what  number? 

50.  -£  of  54  is  i^s  of  what  number  ? 

EXERCISES    FOR    THE    SLATE. 

296.  It  will  be  seen  from  the  preceding  examples, 
that  no  particular  rules  can  be  prescribed  for  solving 
questions  by  analysis.  None  in  fact  are  requisite.  The 
process  will  be  easily  suggested  by  the  judgment  of  the 
pupil,  and  the  conditions  of  the  question. 

OBS.  The  operation  of  solving  a  question  by  analysis,  is  called  an 
analytic  solution.  In  reciting  the  following  examples,  the  pupil 


QITEST.— •?%.  Can  any  particular  rules  be  prescribed  for  solving 
•uestions  by  analysis  ?  How  then  will  you  know  how  to  proceed! 
Obs.  What  is  the  operatian  of  solving  questions  by  analysis,  called  ? 


278  ANALYSIS.  [SECT. 

should  be  required  to  analyze  each  question,  and  give  the  reason  ibr 
each  step,  as  in  the  preceding  mental  exercises. 

Ex.  1.  If  40  barrels  of  beef  cost  $320,  how  much  will 
52  barrels  cost  ? 

Analytic  Solution. — Since  40  bbls.  cost  $320,  1  bbl, 
will  cost  -fa  of  $320.  And  -fa  of  $320  is  $320-^40^$8. 
Now  if  1  bbl.  cost  $8,  52  bbls.  will  cost  52  times  as  much  ; 
and  $8x52— $416,  which  is  the  answer  required. 

Or  thus :  52  bbls.  are  ^  of  40  bbls. ;  therefore  52  bbls 
will  cost  -f-2-  of  $320 ;  (the  cost  of  40  bbls.  ;)  and  -f-£  o* 
$320  is  $320x-f-tr=$416,thesame  result  as  before.  (Arts 
132,  133.) 

OBS. — 1.  Other  solutions  of  this  example  might  be  given;  but  our 
present  object  is  to  show  how  this  and  similar  examples  may  be  solved 
I  y  analysis.  The  former  method  is  the  simplest  and  most  strictly 
analytic,  though  not  so  short  as  the  latter.  It  contains  two  steps: 

First,  we  separate  the  given  price  of  40  bbls.  ($320  )  into  40  equal 
parts,  to  find  the  value  of  one  part,  or  the  cost  of  1  bbl.,  which  is  $8. 

Second,  we  multiply  the  price  of  1  bbl.  ($8  )  by  52,  the  number  oi 
barrels,  whose  cost  is  required,  and  the  product  is  the  answer  sought. 

2.  In  solving  questions  analytically,  it  may  be  remarked  in  general, 
that  we  reason  from  the  given  number  to  1 ,  then  from  1  to  the  re- 
quired number. 

3.  This  and  similar  questions  are  usually  placed  under  Simple  Pro- 
portion, or  the  "  Rule  of  Three;"  but  business  men  almost  invariably 
solve  them  by  analysis. 

2.  If  30  cows  cost  $360.90,  how  much  will  47  cows 
cost,  at  the  same  rate? 

3.  If  25  barrels  of  apples  cost  $15,  how  much  will  37 
barrels  cost  ? 

4.  If  15  hogsheads  of  molasses  cost  $450,  how  much 
will  21  hogsheads  cost? 

5.  If  31  yards  of  cloth  cost  $127,  how  mucn  will  89 
yards  cost  ? 

6.  If  55  tons  of  hay  cost  $660,    what  will   17  tons 
come  to  ? 

7.  An  agent  paid  $159  for  530  pounds  of  wool :  how 
much  was  that  per  100? 

8.  A  man  bought  30  cords  of  wood  for  $76.80 :  how 
much  must  he  pay  for  65  cords? 


A.RT.  296.]  ANALYSIS.  279 

9.  A  gentleman  bought  85   yards   of  carpeting   for 
$106.25  :  how  much  would  38  yards  cost  *?  • 

10.  A  drover  bought  350  sheep  for  $525 :  how  much 
would  65  cost,  at  the  same  rate  ? 

11.  If  12-|  pounds  of  coffee  cost  $1.25,  how  much  will 
45  pounds  cost  ? 

12.  If  16£  bushels  of  corn  are  worth  $8,  how  much 
are  25  bushels  worth  ? 

13.  Paid  $20  for  60  pounds  of  tea :  how  much  would 
12£  pounds  cost,  at  the  same  rate  ? 

14.  Bought  41  yards  of  flannel  for  $16.40:  how  much 
would  8f  yards  cost  1 

15.  Bought  18  pounds  of  ginger  for  84.50:  how  much 
will  lOf  pounds  cost? 

16.  If  a  stage  goes  84  miles  in  12  hours,  how  far  will 
it  go  in  15-^  hours? 

17.  If  8  horses  eat  36  bushels  of  oats  in  a  week,  how 
many  bushels  will  25  horses  eat  in  the  same  time  ? 

18.  If  a  Railroad  car  runs  120  miles  in  5  hours,  how 
far  will  it  run  in  1 2-f-  hours  ? 

19.  If  a  steamboat  goes  180  miles  in  1 2  hours,  how  far 
will  it  go  in  5-f  hours  ? 

20.  If  4  men  can  do  a  job  of  work  in  12  days,  how 
long  will  it  take  6  men  to  do  it  ? 

Solution — Since  the  job  requires  4  men  12  days,  it  will 
require  1  man  4  times  as  long;  and  4  times  12  days  are 
48  days.  Again,  it  requires  1  man  48  days,  it  will  re- 
quire 6  men  £  as  long  ;  and  48  days-^-6=8  days,  which 
is  the  answer  required. 

21.  If  6  men  eat  a  barrel  of  flour  in  24  days,  how  long 
will  it  last  10  men? 

22.  If  a  given  quantity  of  corn  lasts  9  horses  96  days, 
how  long  will  the  same  quantity  last  15  horses  ? 

23.  If  12  men  can  build  a  house  in  90  days,  how  long 
will  it  take  20  men  to  build  it  ? 

24.  If  100  barrels  of  pork  last  a  crew  of  20  men  45 
months,  how  long  will  it  last  a  crew  of  28  men  ? 

25.  If  4  stacks  of  hay  will   keep  60  cattle    120  days, 
how  long  will  they  keep  25  cattle  ? 


280  ANALYSIS.  [SECT. 

26.  If  -f  of  a  bushel  of  wheat  cost  30  cents,  what  will 
^  of  a  bushel  cost  ? 

27.  If  -f-  of  a  ton  of  hay  cost  $7,  what  will  £  of  a  ton 
cost? 

28.  If  £  of  a  pound  of  imperial  tea  cost  27  cents,  how 
much  will  £  of  a  pound  cost  ? 

29.  If  f  of  a  ton  of  coal  cost  $2.61,  how  much  will 
f  of  a  ton  cost  ? 

30.  If  -f  of  a  yard  of  silk  cost  6  shillings,  how  much 
will  -f  of  a  yard  cost  ? 

Solution. — Since  f  of  a  yard  cost  6s.,  -J-  will  cost  3s.. 
and  -f  or  1  yard  will  cost  9s.  Again,  if  1  yard  costs  9s.,' 
\  yd.  will  cost  1^-s. ;  and  •£•  yd.  will  cost  7-J  shillings,  which 
is  the  answer  required.  -w 

31.  If  f  of  a  cord  of  wood  cost  $1.80,  how  much  will 
•f-  of  a  cord  cost  ? 

32.  If  f  of  a  yard  of  broadcloth  cost  14  shillings,  how 
much  will  ^  of  a  yard  cost  ? 

33.  A  man  bought  -f-  of  an  acre  of  land  for  $56,  and 
afterwards  sold  -jf-  of  an  acre  at  cost :  how  much  did  he 
receive  for  it  ? 

34.  A  grocer  bought  7  barrels  of  vinegar  for  $28,  and 
sold  -f-  of  a  barrel  at  cost :  how  much  did  it  come  to  ? 

35.  A  grocer  bought  a  firkin  of  butter  containing  56 
Ibs.  for  $1 1.20,  and  sold  ^  of  it  at  cost :   how  much  did 
he  get  a  pound? 

36.  If  6-J-  bushels  of  peas  are  worth  $5. 50,  how  much 
are  20^-  bushels  worth  ? 

37.  If  a  man  pays  $47  for  building  23£  rods  of  orna- 
mental fence,  how  much  would  it  cost  him  to  build   42^ 
rods? 

38.  A  farmer   paid  $45.42   for  making  36f  rods  o 
stone  wall :  how  much  will  it  cost  him  to  make  60-^- 
rods  ? 

39.  A  man  paid  -nya<r  of  a  dollar  for  4  pounds  of  veal 
how  much  would  a  quarter  of  veal  cost,  which  weighs 
20  pounds  ? 

40.  If  5  pounds  of  butter  cost  4f  shillings,  how  much 
ivill  42  pounds  cost  ? 


296.]  ANALYSIS.  281 

Arofe.— It  will  be  seen  that  4fs.  =-^s.  (Art.  122.)  Therefore  1 
yound  will  cost  f s. ;  and  42  Ibs.  \vill  cost  f  X42-='ifal  or  36s.  Am 

41.  If  20  pounds  of  cheese  cost  $3f,  how  much  will 
168  pounds  cost? 

42.  If  30  yards  of  cot'on  cost  $4|,  how  much  will  a 
piece  containing  19  yards  cost  ? 

43.  If  T\  of  a  cord  of  wood  costs  -£  of  a  dollar,  how 
imch  will  f  of  a  cord  cost  ? 

Solution. — Since  -^  of  a  cord  cost  &£,  -fa  will  cost  $4- ; 
consequently  -^f  or  1  cord  will  cost  S1^.  Again,  if  1 
cord  cost  S-1^,  -£•  of  a  cord  will  cost  $f ;  and  f  will  there- 
fore cost  -f  or  $!-£-,  which  is  the  answer  required. 

44.  If -f-  of  a  yard  of  cloth  cost  £f,  how  much  will  f  of 
a  yard  cost  ? 

"45.  If  -ft-  of  a  ship  cost  $16000,  how  much  is  £  of  her 
worth  ? 

46.  A  man  bougkt  a  quantity  of  land  and  sold  -fa  of  an 
acre  for  $63,  which  was  only  f  of  the  cost :  how  much 
did  he  give  per  acre  ? 

47.  If  1\  yards  of  satinet  cost  $9-f ,  how  much  wiL 
1 8-£  yards  cost  ? 

48.  A  ship's  company  of  30  men  have  4500  pounds  of 
flour:  how  long  will  it  last  them,  allowing  each  man  2£ 
Ibs.  per  day  ? 

49.  How  long  will  56700  pounds  of  meat  last  a  garri- 
son of  756,  allowing  each  man  f  Ib.  per  day  ? 

50.  How  long  will  the  same  quantity  of  meat  last  the 
same  garrison,  allowing  \\  Ib.  apiece  per  day  1 

51.  A  merchant  sold  12  yards  of  silk  at  7  shillings  per 
yard,  and  took  his  pay  in  wheat  at  6  shillings  per  bushel . 
how  many  bushels  did  he  receive  ? 

Solution. — First  find  the  cost  of  the  silk.  If  1  yd.  costs 
7s.,  12  yds.  will  cost  7  times  12s.,  which  is  84s."  Now 
the  question  is,  how  many  bushels  of  wheat  it  will  take 
to  pay  this  84s.  But  as  the  wheat  is  6s.  a  bushel,  it  will 
manifestly  take  as  many  bushels  as  6s.  is  contained  times 
in  84s.;  and  84-*-6«14.  Am.  14  bushels. 


282  ANALYSIS.  [SECT. 

297.  The  last  and  simi.ar  examples  are  frequently 
solved  by  a  rule  called  Barter. 

Barter  signifies  an  exchange  of  articles  of  commerce 
at  prices  agreed  upon  by  the  parties. 

OBS.  A  specific  rule  for  such  operations  seems  to  be  worse  than 
useless;  for  it  burdens  the  memory  of  the  learner  with  particular  di- 
rections for  the  solution  of  questions  which  his  common  sense,  if  per- 
mitted to  be  exercised,  will  solve  more  expeditiously  by  the  principle* 
of  analysis. 

52.  A  shoemaker  sold  6  pair  of  thick  boots  at  32  shii 
lings  a  pair,  and  took  his  pay  in  corn  at  3  shillings  peJ\ 
oushel :  how  many  bushels  did  he  receive? 

53.  A  marl  bought  50  pounds  of  sugar  at  12-J-  cent" 
a  pound,  and  was  to  pay  for  it  in   wood  at  $3.12-£  per 
cord  :  how  many  cords  did  it  take  ? 

54.  How  many  pair  of  hose  at  3  shillings  a  pair,  will 
it  take  to  pay  for  135  pounds  of  tea  at  6  shillings  a  pound  ? 

55.  How  many  pounds  of  butter  at  17-J-  cents  a  pound, 
must  be  given  in  exchange  for  186  yards  of  calico  at  18-^ 
cents  per  yard  ? 

56.  How  many  pounds  of  tobacco  at  16-J-  cents  a  pound, 
must  be  given  in  exchange  for  256  pounds  of  sugar  at  6^ 
cents  a  pound  ? 

57.  A  farmer  bought  325  sheep  at  $2-£  apiece,  and 
paid  for  them  in  hay  at  $10£  per  ton  :  how  many  tons 
did  it  take  ? 

58.  A  man  bought  a  hogshead  of  molasses  worth  37  J 
cents  per  gallon,  and  gave  33 1£  pounds  of  cheese  in  ex- 
change :  how  much  was  the  cheese  a  pound  ? 

59.  Bought  74  bushels  of  salt  at  42^-  cents  per  bushel, 
and  paid  in  oats  at  |  of  a  dollar  per  bushel :  how  many 
oats  did  it  require  ? 

60.  A  bookselier  exchanges  400  dictionaries  worth  87-J 
cents  apiece  for  700  grammars :  how  much  did  the  gram 
mars  cost  apiece  ? 

61.  What  cost  680  tons  of  chalk,  at  10  shillings   ster 
ling  per  ton? 

QUEST. —297.  What  is  meant  by  Barter  ?  O6*.  Is  a  specific  r«U 
necessary  for  such  operations  * 


X 

.gdjF   ARTS 


.297,298.] 


ANALYSIS. 


283 


Solution.— 10s.  =  £i.  Now,  if  1  ton  costs  ££,  680  tons 
will  cost  (380  times  as  much ;  that  is,  680  tons  will  cost 
half  as  many  pounds  as  there  are  tons  ;  (Art.  132  ;)  and 
680-r2  =  340.  Arts.  £340. 

29§.  Examples  like  the  preceding  one  are  often 
classed  under  the  Rule  called  Practice. 

TABLE    OF    ALIQUOT    TARTS    OF    $1,    £l,    AND    Is. 


Parts  of  $1. 


N.  E.  Currency 


3  shil.=8i. 
2  shil.=$i. 
Is.  6d.=$i. 
1  shil.  =  Si. 


N.  Y.  Currency 

4  shil.=$i. 
2s.  8d.=§i. 
2  shil.  =  8|  . 
Is.  4d.=Sf 


Parts  of  jei. 


10  shil.=£i. 
6s.  8d.  =££. 

5  shil.  =  £i. 

4  shil.=£i. 
3s.  4d.  =  £i. 
2s.  6d.  =  ££. 

2  shil. 
Is.  8d.  = 
1  shil.   = 


Parts  of  Is. 


=ishil. 
=  i  shil. 
=ishil. 


6d. 
4d. 
3d. 
2d. 

d.=  ishil. 
Id.   =-jVshil. 
id.   =2Vshil. 
=f  shil. 


9d. 


8d.   =     shil. 


62.  What  cost  720  bushels  of  corn,  at  2  shillings  and 
6  pence  per  bushel  ? 

Solution.— 2s.  6d.  =  £i.     And  720Xi  =  £90.  Ans. 

63.  What  cost  840  chairs,  at  3s.  N.  E.  cur.  apiece/ 

64.  What  cost  360  melons,  at  Is.  6d.  N.  E.  cur.  apiece? 

65.  What  cost  360  knives,  at  2s.  8d.  N.  Y.  cur.  apiece  ? 

66.  What  cost  760  brooms,  at  Is.  N.  Y.  cur.  apiece? 

67.  At  10s.  6d.  sterling  per  barrel,  what  will  350  bar- 
rels of  mackerel  come  to  ? 

68.  At  17s.  6d.  st.  apiece,  what  will  540  hats  come  to  ? 

69.  What  cost  33750  sheep,  at  6s.  8d.  st.  apiece  ? 

70.  At  $20.60  per  ton,  what  cost  12  tons  and  5  hun- 
dred weight  of  hay  ? 

71.  What  is  the  cost  of  480  yards  of  ribbon,  at  6-J 
cents  per  yard  ? 

72.  What  cost  750  bushels  of  potatoes,  at  33£  cents 
per  bushel  ? 

73.  What  cost  360  barrels  of  cider,  at  66|  cents  per 
barrel  ? 


284  ANALYSIS.  [SECT. 

74.  What  cost  450  chaldrons  of  coal,  at  15s.  per  chal« 
dron? 

75.  What  will  150  acres  of  land  cost,  at  £8,   10s.  pei 
acre? 

76.  Three  men,  A,  B.  and  C  join  in  an  adventuie  ;  A 
puts  in   $200 ;  B,  $300 ;  and  G,  $400  ;  and  they  gain 
$72 :  how  much  is  each  man's  share  of  the  gain  ? 

Analysis. — The  whole  sum  invested  is  S200+$300-h 
$400=$900.     Now,  since  $900  gain  $72,  $1  will  gain 
-y^Tf  of  $72  ;  and  $72-s-900=$.08. 
If  $1  gains  8c.,  $200  will  gain  $200x.08=$16,  A's  sh. 
«      1      «       «         300         «  300x.08=   24,  B's  " 

«      1      "     .  «         400         "  400x.08^   32,  C's  " 

Or,  we  may  reason  thus  :  since  the  sum  invested  is  $900 
A's  part  of  the  investment  is  -££|j-=f ; 
B's         "  *'•          isW=f; 

C's         "  isW=i-     Hence, 

A  must  receive  -f-  of  $72  (the  gain)=$16 
B          "  -f"      72         «         =   24 

C          "  i"      72         «         =_32 

PROOF.— The  whole  gain  is  $72.     (Ax.  11.) 

29  9»  When  two  or  more  individuals  associate  them 
selves  together  for  the  purpose  of  carrying  on  a  joini 
business,  the  union  is  called  a  partnership  or  copartnership 

OBS.  The  process  by  which  examples  like  the  last  one  are  solved, 
is  often  called  Fellowship. 

77.  A  and  B  entered  into  partnership ;  A  furnished 
$400,  and  B  $500  ;  they  gained  $300 :  how  much  was 
each  man's  share  of  the  gain  ? 

78.  A,  B,  and  C  hired  a  farm  together,  for  which  they 
paid  $175  rent :  A  advanced  $75  ;  B,  $60  ;  and  C,  $40. 
They  raised  250  bushels  of  wheat :  what  was  each  man's 
share  ? 

79.  A,  B,  and  C  together  spent  $1000  in  lottery  ticketa 
A  put  in  $400 ;  B,  $250 ;  and  C,  $350 ;  they  drew  a 
prize  of  $  1 500  :  how  much  wag  each  man's  share  ? 


ARTS  299-30 l.J  ANALYSIS.  285 

80.  A,  B,  C,  and  D  fitted  out  a  whale  ship ;  A  advan- 
ced $10000;  B,  $12000;  C,  $15000;  and  D,  $8000; 
the  ship  brought  home  3000  bbls.  of  oil :  what  was  each 
man's  share  ? 

81.  A,  B,  and  C  formed  a  partnership;  A  furnished 
$900;    B,  $1500;    and  C,   $1200:    they    lost  $1260; 
what  was  each  man's  share  of  the  loss? 

82.  X,  Y,  and  Z  entered  into  a  joint  speculation,  on  a 
capital  of  $20000.    of  which   X  furnished  $5000 ;   Y, 
$7000  ;  and  Z  the  balance  ;  their  net  profits  were  $5000 
per  annum :  what  was  the  share  of  each  ? 

83.  A  bankrupt  owes  one  of  his  creditors  $300 ;  an- 
other $400  ;  and  a  third  $500  ;  his  property  amounts  to 
$800  :  how  much  can  he  pay  on  a  dollar  ;  and  how  much 
will  each  of  his  creditors  receive? 

Note. — The  solution  of  this  example  is  the  same  in  principle  as 
example  seventy-sixth. 

300.  A  bankrupt  is  a  person  who  is  insolvent,  or 
unable  to  pay  his  just  debts. 

OBS.  Examples  like  the  preceding  one  are  sometimes  arranged  un- 
der a  rule  called  Bankruptcy. 

84.  A  bankrupt  owes  $2000,  and  his  property  is  ap- 
praised at  $1600  :  how  much  can  he  pay  on  a  dollar  ? 

85.  A  man  failing  in  business,  owes  A  $156.45;  B 
$256.40 ;  and  C  $360.40  ;  and  his  effects  are  valued  at 
$317 :  how  much  will  each  man  receive  ? 

86.  The  whole  effects  of  a  man  failing   in  business 
amounted  to  $3560,  he  owed  $35600  :  how  much  can  he 
pay  on  a  dollar ;  and  how  much  will  B  receive,  who  has 
a  claim  on  him  of  $5000  ? 

87.  A  man  died  insolvent,   owing  $55645 ;  and  his 
oroperty  was  sold  at  auction  for  $2350  :  how  much  will 
his  estate  pay  on  a  dollar  ? 

88.  How  much  can  a  bankrupt,  who  has  $6540  real 
estate  and  owes  $56000,  pay  on  a  dollar  ? 

301.  It  often  happens  in  storms  and  other  casualties 
at  soa,  that  masters  of  vessels  are  obliged  to  throw  par- 


286  ANALYSIS.  [SECT.   X! 

tions  of  their  cargo  overboard,  or  sacrifice  the  ship  and 
their  crew.  In  such  cases,  the  law  requires  that  the  loss 
shall  be  divided  among  the  owners  of  the  vessel  and  cargo, 
in  proportion  to  the  amount  of  each  one's  property  at 
stake. 

The  process  of  finding  each  man's  loss,  in  such  instan- 
ces, is  called  General  Average. 

OBS.  The  operation  is  the  same  as  that  in  solving  questions  in  bank- 
ruptcy and  partnership. 

89.  A,  B,  and  C  freighted  a  sloop  with  flour  from  New 
York  to  Boston  ;  A  had  on  board  600  barrels  ;  B,  400  ; 
an^  J,  xiOO.     On  her  passage  200  barrels  were  thrown 
overboard  in  a  gale,  and  the  loss  was  shared  among  the 
owners  according  to  the  quantity  of  flour  each  had  on 
board :  what  was  the  loss  of  each  ? 

90.  A  Liverpool  packet  being  in  distress,  the  master 
threw  goods  overboard  to  the  amount  of  $10000.     The 
whole  cargo   was  valued   at   $72000,  and  the  ship   at 
$28000  :  what  per  cent,  loss  was  the  general  average ; 
and  how  much  was  A's  loss,  who  had  goods  aboard  to 
the  amount  of  $15000? 

91.  A  coasting  vessel  being  overtaken  in  a  gale,  the 
master  was  obliged  to  throw  overboard  part  of  his  cargo 
valued  at  $15500.     The  whole  cargo  was  worth  $85265, 
and  the  vessel  $17000:  what  per  cent,  was  the  general 
average  ;  and  what  was  the  loss  of  the  master,  who  owned 
$  of  the  vessel  ? 

92.  A  farmer  mixed  15  bushels  of  oats  worth  2  shil- 
lings per  bushel,  with  5  bushels  of  corn  worth  4  shillings 
per  bushel :  what  is  the  mixture  worth  per  bushel  1 

Solution. — 15  bu.  at  2s.=30s.,  value  of  oats. 
5  bu.  at  4s.=20s.,  value  of  corn. 
20  bu.  mixed.  50s.,  value  of  whole  mixture. 

Now,  if  20  bu.  mixture  are  worth  50s.,  1  bu.  is  worth 
-£r  of  50s.,  which  is  2£s.,  the  answer  required. 

PROOF. — 20  bu,x2-£&.=50s.   the  value  of  the   whole 
mixture. 


,RT.   302.]  ANALYSIS.  287 

93.  A  miller  has  a  quantity  of  rye  worth  6s.  per  bushel, 
and  wheat  worth  9s.  per  bushel ;  he  wishes  to  make  a 
mixture  of  them  which  shall  be  worth  8s.  per  bushel . 
what  part  of  each  must  the  mixture  contain  ? 

Analysis. — The  difference  in  their  prices  per  bushel  is 
3s. ;  hence,  the  difference  in  the  price  of  1  third  of  a 
bushel  of  each  is  Is.  Now  if  1  third  of  a  bushel  is  taken 
from  a  bushel  of  rye,  the  remaining  2  thirds  will  be 
worth  4s. ;  and  if  1  third  of  a  bushel  of  wheat  which  is 
worth  3s.,  be  added  to  the  rye,  the  mixture  will  be  worth 
Vs.  Again,  if  f  of  a  bushel  is  taken  from  a  bushel  of  rye, 
the  remaining  third  will  be  worth  2s.,  and  if  -f  of  a  bushel 
of  wheat,  which  is  worth  6s.,  be  added  to  the  rye,  the 
mixture  will  be  worth  8s. ;  therefore,  -J-  of  a  bushel  of  rye 
added  to  -f  of  wheat,  will  make  a  mixture  of  1  bushel, 
which  is  worth  8  shillings ;  consequently  the  mixture 
must  be  %  rye  and  f  wheat ;  or  1  part  rye  to  2  parts 
wheat. 

PROOF. — Since  1  bushel  of  rye  is  worth  6s.,  •£  bu.  is 
worth  -£  of  6s.,  or  2s. ;  and  as  1  bu.  of  wheat  is  worth  9s., 
|  bu.  is  worth  f  of  9s.,  or  6s. ;  and  6s.-f2s.=8s. 

*ti 

Note. — If  we  make  the  difference  between  the  less  price  and  the 
price  of  the  mixture,  the  numerator,  and  the  difference  betweeen  the 
prices  of  the  commodities  to  be  mixed,  the  denominator,  the  fraction 
will  express  the  part  to  be  taken  of  the  higher  priced  article ;  and  if 
we  place  the  difference  between  the  higher  price  and  the  price  of  the 
mixture  over  the  same  denominator,  the  fraction  will  express  the  part 
to  betaken  of  the  lower  priced  article. 

94.  A  goldsmith  has  a  quantity  of  gold  16  carats  fine, 
and  another  quantity  22  carats  fine  ;  he  wishes  to  make  a 
mixture  20  carats  fine  :  what  part  of  each  will  the  mixture 
contain  ? 

Ans.  f  of  16  carats  fine,  and  £  of  22  carats  fine. 

3 O2»  Examples  requiring  a  mixture  of  commodities 
of  different  values,  like  the  last  three,  are  commonly 
:lassed  under  a  rule  called  Alligation. 

Alligation  is  usually  divided  into  mtctial  and  alt»mat&    Th» 


288  ANALYSIS.  [$Eo 

92d  example  is  an  instance  of  Medial  Alligation ;  the  93d  and  94th 
are  instances  of  Alternate  Alligation.  Questions  in  the  latter  very 
neldom  occur  in  practical  life. 

95.  A  grocer  mixes  50  pounds  of  tea  worth  4  shillings 
a  pound,  with   100  Ibs.  worth   7s.  a  pound  :  what  is  a 
pound  of  the  mixture  worth  ? 

96.  A  milk-man  mixed  30  quarts  of  water  with  120 
quarts  of  milk,  worth  5  cents  per  quart :  what  is  a  quart 
of  the  mixture  worth  1 

97.  A  farmer  made  a  mixture  of  provender  containing 
30  bushels  of  oats,  worth  25  cents  per  bushel ;   10  bushels 
of  peas,  worth  75   cents  per  bushel,  and   15  bushels  oi 
corn,  worth   50  cents  per  bushel :  what  is  the  value  of 
the  whole  mixture  ;  and  what  is  it  worth  per  bushel  ? 

98.  An  oil  dealer  mixed  60  gallons  of  whale  oil,  worth 
31-J-  cents  per  gallon,  with  85  gallons  of  sperm  oil,  worth 
90  cents  per  gallon:    what   is   the  mixture  worth  per 
gallon  ? 

99.  A  grocer  had  three  kinds  of  sugar,  worth  6,  8,  and 
12  cents  per  pound  ;  he  mixed  112  Ibs.  of  the  first,  150 
Ibs.  of  the  second,  and  175  of  the  third  together:   what 
was  the  mixture  worth  per  pound  ? 

100.  A  goldsmith  melted  10  oz.  of  gold  20  carats  fine, 
with  8  oz.  22  carats  fine,  and  4  oz.  of  alloy :  how  many 
carats  fine  was  the  mixture  ? 

101.  If  4  men  reap  12  acres  in  2  days,  how  long  will 
it  take  9  men  to  reap  36  acres  ? 

Analysis. — If  4  men  can  reap  12  acres  in  2  days,  1 
man  can  reap  \  of  12  acres  in  the  same  time;  and  \  of 
12  acres  is  3  acres.  But  if  1  man  can  reap  3  acres  in 
2  days,  in  1  day  he  can  reap  i  of  3  acres,  and  £  of  3  is  1 J- 
acre.  Again,  if  1-J-  acre  requires  a  man  1  day,  36  acres 
will  require  him  as  many  days  as  1£  is  contained  times 
in  36  ;  and  36-*-l-J-=24  days.  Now  if  1  man  can  reap 
the  given  field  in  24  days,  9  men  will  reap  it  in  •£  of  the 
time :  and  24-s-9=2|. 

Am.  9  men  can  reap  36  acres  in  2-f  days, 

OBR. — This  and  similar  examples  are  usually  placed  under  Com- 
pound Proportion,  or  "Double  Rule  of  Three/'  If  the  analysis  of 


.  303.]  ANALYSIS.  289 

them  is  found  too  difficult  for  beginners,  they  can  he  deferred  tiH 
.eview. 

102.  If  7  men  can  reap  42  acres  in  6  days,  how  many 
men  will  it  take  to  reap  100  acres  in  5  days  ? 

103.  If  14   men  can  build  84  rods  of  wall  in  3  days, 
how  long  will  it  take  20  men  to  build  300  rods? 

104.  If  1000  barrels  of  provisions  will  support  a  garri- 
son of  75  men  for  3  months,  how  long  will  3000  barrels 
support  a  garrison  of  300? 

105.  If  a  man  travels  320  miles  in   10  days,  traveling 
8  hours  per  day,  how  far  can  he  go  in  15  days,  traveling 
12  hours  per  day? 

106.  If  24  horses  eat  126  bushels  "of  oats  in  36  days, 
how  many  bushels  will  32  horses  eat  in  48  days  ? 

107.  A  lad  returning  from  market  being  asked  how 
many  peaches  he  had  in  his  basket,  replied  that  •£,  -J-,  and 
%  of  them  made  52  :  how  many  peaches  had  he '/ 

Analysis.— The  sum  of  £,  |,  and  -J-=ff.  (Art.  127.) 
The  question  then  resolves  itself  into  this :  52  is  -}-f  of 
what  number  ?  Now  if  52  is  -ff,  -fa  is  -fa  of  52,  which 
is  4  ;  and  ±Z  is  4x12=48.  Ans.  48  peaches. 

PROOF.—- £  of  48  is  24 ;  i  is  16 ;  and  i  is  12.  Now, 
24+16+12=52. 

3O3.  This  and  similar  examples  are  often  placed 
under  a  rule  called  Position. 

OBS.  The  shortest  and  easiest  method  of  solving  them  is  by  Anal- 
ysis. 

108.  A  farmer  lost  •£  of  his  sheep  by  sickness  ;  %  were 
destroyed  by  wolves ;  and  he  had  72  sheep  left :  how 
many  had  he  at  first  ? 

109.  A  person  having  spent  $  and  %  of  his  money, 
finds  he  has  $48  left:  what  had  he  at  first? 

110.  After  a  battle  a  general  found  that  •§•  of  his  army- 
had  been  taken  prisoners,  \  were  killed,  -fa  had  deserted, 
and  he  had  900  left ;  how  many  had  he  at  the  commence- 
ment of  the  action  ? 

10 


290  ANALYSIS.  [SECT.    XII 

111.  What  number  is  that  •£  and  |  of  which  is  84 1 

112.  What  number  is  that  •£  and  •£  of  which  being 
added  to  itself,  the  sum  will  be  110? 

113.  A  certain    post  stands  -J-  in  the   mud,  \  in  the 
water,  and  10  feet  above  the  water:  how  long  is  the  post? 

114.  Suppose  I  pay  $85  for  -f  of  an  acre  of  land: 
what  is  that  per  acre  ? 

1 15.  A  man  paid  $2700  for  -^  of  a  vessel :  what  is  the 
whole  vessel  worth? 

116.  A  gentleman  spent  •£•  of  his  life  in  Boston,  •}•  of  it 
in  New  York,  and  the  rest  of  it,  which  was  30  years,  in 
Philadelphia: how  old  was  he? 

"*  1 1 7.  What  number  is  that  •£  of  which  exceeds  -f  of  it 
by  10? 

118.  In  a  certain  school  -£•  of  the  scholars  were  studying 
arithmetic,  -J-  algebra,  •£•  geometry,  and  the  remainder, 
which  was   18,   were   studying  grammar:    how   many 
scholars  were  there  in  the  school  ? 

1 19.  A  owns  i,  and  B  -^  of  a  ship  ;  A's  part  is  worth 
$650  more  than  B's:  what  is  the  value  of  the  ship? 

120.  In  a  certain  orchard  %  are  apple-trees,  -J-  peach 
trees,  -J-  plumb-trees,  and  the  remaining  15  were  cherry 
trees  :  how  many  trees  did  the  orchard  contain  ? 


SECTION  XII. 
RATIO  AND  PROPORTION. 

ART.  3O5.  RATIO  is  that  relation  between  two  num- 
bers or  quantities,  which  is  expressed  by  the  quotient  ol 
the  one  divided  by  the  Other.  Thus,  the  ratio  of  6  to  2 
is  G-i-2,  or  3 ;  for  3  is  the  quotient  of  6  divided  by  2. 

MENTAL    EXERCISES. 

£x.  1.  What  is  the  ratio  of  14  to  7?  Ans,  2. 

2.  What  is  the  ratio  of  10  to  2  ?     Of  16  to  4? 

QUEST.— 305.  What  1«  ratio  1 


jf/Ers.  305-307.]  -RATIO.  29, 

3.  What  is  the  ratio  of  18  to  9  ?     Of  18  to  6  ? 

4.  What  is  the  ratio  of  24  to  3  ?     Of  24  to  4  ?     Of  24 
to 6?     Of  24  to  8?     Of  24  to  12? 

5.  What  is  the  ratio  of  30  to  6  ?     Of  25  to  5  ?     Of  27 
to  9?     Of  40  to  8?     Of  56  to  7?     Of  84  to  12? 

G.  WThat  is  the  ratio  of  3  to  7  ?  Ans.  f. 

7.  What  is  the  ratio  of  5  to  8  ?  Of  7  to  10  ?  Of  9 
to  13?  Of  10  to  17?  Of  21  to  43? 

306.  The  two  given  numbers  thus  compared,  when 
spoken  of  together,  are  called  a  couplet ;  when  spoken  of 
separately,  they  are  called  the  terms  of  the  ratio. 

The  first  term   is   the  antecedent ;   and  the   last,  the 

consequent. 

307.  Ratio  is  expressed  in  two  ways: 

First,  in  the  form  of  a  fraction,  making  the  antecedent 
the  numerator,  and  the  consequent  the  denominator.  Thus, 
the  ratio  of  8  to  4  is  written  -f- ;  the  ratio  of  12  to  3,  ~2-,  &c. 

Second,  by  placing  two  points  or  a  colon  ( :  )  between 
the  numbers  compared.  Thus,  the  ratio  of  8  to  4,  is 
written  8:4;  the  ratio  of  12  to  3,  12  :  3,  &c. 

OBS.  1.  The  expressions  i",  and  8  :  4  are  equivalent  to  each  other, 
and  one  may  be  exchanged  for  the  other  at  pleasure. 

2.  The  English  mathematicians  put  the  antecedent  for  the  nume- 
rator and  the  consequent  for  the  denominator,  as   above;  but  the 
French  put  the  consequent  for  the  numerator  and  the  antecedent  for 
the  denominator.     The  English  method  appears  to  be  equally  simple, 
and  is  confessedly  the  most  in  accordance  with  reason. 

3.  In  order  that  concrete  numbers  may  have  a  ratio  to  each  other, 
they  must  necessarily  express  objects  so  far  of  the  same  nature,  that 
one  can  be  properly" said  to  be  equal  to,  or  gr cater ,  or  less  than  the 
other.  (Art.  280.)     Thus  a  foot  has  a  ratio  to  a  yard;  for  one  is  three 
times  as  long  as  the  other;  but  a  foot  has  not  properly  a  ratio  to  an 
hour,  for  one  cannot  be  said  to  be  longer  or  shorter  than  the  other. 

QUEST. — 306.  What  are  the  two  given  numbers  called  when  spoken 
cf  together  ?  What,  when  spoken  of  separately  ?  307.  In  how  many 
ways  is  ratio  expressed  ?  What  is  the  first  ?  The  second  I  Obs.  Which 
of  the  terms  do  the  English  mathematicians  put  for  the  numerator  I 
Which  do  the  French  ?  In  order  that  concrete  numbers  may  have  f 
ratio  to  «ach  other,  what  kind  of  objects  must  they  express  ? 


ILgk 

292  RATIO.  [SECT.  3&V 

% 

8.  What  is  the  ratio  of  15  Ibs.  to  3  Ibs.  ?     Of  21  Ibs.  td  * 
7  Ibs.  ?     Of  35  bu.  to  7  bu.  ?     Of  36  yds.  to  12  yds  ? 

9.  What  is  the  ratio  of  £1  to  10s.  1 

Notc.—£l  is  20s.  The  question  then  is  simply  this:  what  is  thff 
ratio  of  20s.  to  10s.  3  Ans.  2. 

10.  What  is  the  ratio  of  £2  to  5s.  ?     Of  £3  to  12s.  ? 

308.  A  direct  ratio  is  that  which  arises  from  dividing 
iiie  antecedent  by  the  consequent,  as  in  Art.  305. 

3O9«  An  diverse  or  reciprocal  ratio,  is  the  ratio  of  the 
reciprocals  of  two  numbers.  (Art.  280.  Def.  1 1.)  Thus, 
the  direct  ratio  of  9  to  3.  is  9  :  3,  or  -f-;  the  reciprocal 
ratio  is  -5-  :  i,  or  $-*-£=% ;  (Art.  139  ;)  that  is,  the  conse- 
quent 3,  is  divided  by  the  antecedent  9.  Hence, 

A  reciprocal  r.atio  is  expressed  by  inverting  the  fraction 
which  expresses  the  direct  ratio ;  or  when  the  notation  is  by 
points,  by  inverting  the  order  of  ike  terms.  Thus,  8  is  to 
4,  inversely,  as  4  to  8. 

309.  a.  A  simple  ratio  is  a  ratio  which  has  but  one 
antecedent  and  one  consequent,  and  may  be  either  direct 
or  inverse  ;  as  9  :  3,  or  -J- :  -J-. 

3 1 0.  A  compound  ratio  is  the  ratio  of  the  products  of  the 
corresponding  terms  of  two  or  more  simple  ratios.     Thus, 

The  simple  ratio  of  9  :    3  is  3  ; 

And      "         "of  8  :    4  is  2 ; 

The  ratio  compounded  of  these  is  72  :  12  =  6; 

OBS.  A  compound  ratio  is  of  the  same  nature  as  any  other  ratio. 
The  term  is  used  to  denote  the  origin  of  the  ratio  in  particular  cases - 

311*  From  the  definition  of  ratio  and  the  mode  oi 
expressing  it  in  the  form  of  a  fraction,  it  is  obvious  that 
the  ratio  of  two  numbers  is  the  same  as  the  value  of  a 
fraction  whose  numerator  and  denominator  are  respec- 
tively equal  to  the  antecedent  and  consequent  of  the  giv* 

QUEST.— 308.  What  is  a  direct  ratio  ?  309.  What  is  an  inverse  ot 
reciprocal  ratio  ?  How  is  a  reciprocal  ratio  expressed  by  a  fraction ? 
How  by  points  ?  309.  a.  What  is  a  simple  ratio  ?  310.  What  is  a 
compound  ratio!  Obs.  Does  it  differ  in  its  nature  from  othftr  ratios! 
HI.  Wbftt  k  th«  ratio  of  two  number*  *r*val  tcH 


JETS.  308-315.]  RATIO.  293 

en  couplet ;  for,  each  is  the  quotient  of  the  numerator  di- 
vided by  the  denominator.  (Arts.  305,  110.) 

OBS.  From  the  principles  of  fractions  already  established,  we  may, 
therefore,  deduce  the  following  truths  respecting  ratios. 

312.  To  multiply  the  antecedent  of  a  couplet  by  any 
number,  multiplies  the  ratio  by  that  number  ;  and  to  dzvide 
the  antecedent,  divides  the  ratio:  for,  multiplying  the  nume- 
rator, multiplies  the  value  of  the  fraction  by  that  number, 
and  dividing  the  numerator,  divides  the  value.  (Arts.  1 1 1 
112.) 

Thus,  the  ratio  of         16  :  4  is  4 ; 

The  ratio  of  16x2  :  4  is  8,  which  equals  4x2  ; 

And       "  16-t-2  :  4  is  2,      "          "       4-*-2. 

313*  To  multiply  the  consequent  of  a  couplet  by  any 
number,  divides  the  ratio  by  that  number  ;  and  to  divide  the 
consequent,  multiplies  the  ratio :  for,  multiplying  the  denom- 
inator, divides  the  value  of  the  fraction  by  that  number, 
and  dividing  the  denominator,  multiplies  the  value.  (Arts. 
113,  114.) 

Thus  the  ratio  of  16  :  4       is  4  ; 

The  "  16  :  4x2  is  2,  which  equals  4-^-2  ; 

And  "          16  :  4-s-2  is  8,  which  equals  4x2. 

314.  To  multiply  or  divide  both  the  antecedent  and  con- 
sequent of  a  couplet  by  the  same  number,  does  not  alter  the  ra- 
tio  :  for,  multiplying  or  dividing  both  the  numerator  and 
denominator  by  the  same  number,  does  not  alter  the  value 
of  the  fraction.  (Art.  116.) 

Thus  the  ratio  of       12  :  4       is  3  ; 
The  «          12x2:  4x2  is  3; 

And  "         12-2  : 4-s-2  is  3. 

315.  If  the  two  numbers  compared  are  equal,  the 
ratio  is  a  unit  or  1 :  for,  if  the  numerator  and  denomina- 

QUEST.— 312.  What  is  the  effect  of  multiplying  the  antecedent  of  a 
couplet  by  any  number  ?  Of  dividing  the  antecedent  ?  How  does  this 
appear  ?  313.  What  is  the  effect  of  multiplying  the  consequent  by  any 
number?  Of  dividing  the  consequent?  Why?  314.  What  is  the 
effect  of  multiplying  or  dividing  both  the  antecedent  and  consequent 
by  the  Bame  number  ?  Why  ? 


1 

294  PROPORTION.  [SECT?" 

tor  are  equal,  the  value  of  the  fraction  is  a  unit,  or  1. 
(Art.  117.)  Thus  the  ratio  of  6x2  :  12  is  1  ;  for  the 
value  of  -ff=l.  (Art  121.) 

OBS.  This  is  called  a  ratio  of  equality. 

316.  If  the  antecedent  of  a  couplet  is  greater  thai? 
the  consequent,  the  ratio  is  greater  than  a  unit :  for,  if  the 
numerator  is  greater  than  the  denominator,  the  value  oi 
the  fraction  is  greater  than  1.  (Art.  117.)     Thus  the  ratio 
of  12  :  4  is  3. 

OBS,  This  is  called  a  ratio  of  greater  inequality, 

317.  If  the  antecedent  is  less  than  the  consequent, 
the  ratio  is  less  than  a  unit :  for,  if  the  numerator  is  lest 
than  the  denominator,  the  value  of  the  fraction  is  less 
than  1.  (Art.  117.)     Thus,  the  ratio  of  3  :  6  is  £ ,  or  £; 
for  £=£.  (Art.  120.) 

OBS.  This  is  called  a  ratio  of  less  inequality. 

11.  What  is  the  direct  ratio  of  3  :  9,  expressed  in  the 
lowest  terms  ?     What  the  inverse  ratio  ? 

Ans.  i ;  and  i~H=3.  (Arts.  308,  309.) 

12.  What  is  the  inverse  ratio  of  4  to  12?     Of  6  to  18 ) 
Of  9  to  24  ?     Of  21  to  25  ?     Of  40  to  56  ? 

13.  What  is  the  direct  ratio  of  15s.  to  £2?     Of  13s. 
6d.  to£l?     Of  £2,  10s.  to  £3,  5s.  ? 

14.  What  is  the  direct  ratio  of  6  inches  to  3  feet? 

15.  What  is  the  direct  ratio  of  15  oz.  to  1  cwt.  ? 

PROPORTION. 

318.  PROPORTION  is  an  equality  of  ratios.     Thus,  the 
*wo  ratios  6  :  3  and  4  :  2  form  a   proportion ;  for 

the  ratio  of  each  beinp;  2. 


QUEST. — 315.  When  the  two  numbers  compared  are  equal,  what  if 
the  Batio?  Obs.  What  is  it  called?  316.  When  the  antecedent  it 
greater  than  the  consequent,  what  is  the  ratio  ?  Obs.  What  is  it  call- 
ed? 317.  If  the  antecedent  is  less  than  the  consequent,  what  is  tha 
l»tio  ?  Obs.  What  is  it  called  ?  3 18. -What  is  proportion  ? 


i.  316-320.]  PROPORTION,  295 

OES.  The  terms  of  the  two  couplets,  that  is,  the  numbers  of  which 
Jhe  proportion  is  composed,  are  culled  proportionals. 

319.  Proportion  may  be  expressed  in  two  ways. 

First,  by.  the  sign  of  equality  (=)  placed  between  the 
two  ratios. 

Second,  by  four  points  or  a  double  colon  (:  :)  placed  be- 
tween the  two  ratios. 

Thus,  each  of  the  expressions,  12  :  6—4  :  2,  and 
12  :  6  :  :  4 :  2,  is  a  proportion,  one  being  equivalent  to 
ihe  other. 

OBS.  The  latter  expression  is  read,  "the  ratio  of  12  to  6  equals  the 
ratio  of  4  to  2,"  or  simply,  "  12  is  to  6  as  4  is  to  2." 

3 2O«  The  number  of  terms  in  a  proportion  must  at 
least  be/oM/,  for  the  equality  is  between  the  ratios  of  two 
couplets,  and  each  couplet  must  have  an  antecedent  and  a 
consequent.  (Art.  306.)  There  may,  however,  be  a  pro 
portion  formed  from  three  numbers,  for  one  of  the  numbers 
may  be  repeated  so  as  to  form  two  terms.  Thus  the  num- 
bers 8,  4,  and  2,  are  proportional :  for  the  ratio  of  8  :  4= 
4:2.  It  will  be  seen  that  4  is  the  consequent  in  the  first 
couplet,  and  the  antecedent  in  the  last.  It  is  therefore  a 
mean  proportional  between  8  and  2. 

OBS.  1.  In  this  case,  the  number  repeated  is  called  the  middle  term 
or  mean  proportional  between  the  other  two  numbers. 

The  last  term  is  called  a  third  proportional  to  the  other  two  num- 
bers. Thus  2  is  a  third  proportional  to  8  and  4. 

2.  Care  must  be  taken  not  to  confound  proportion  with  ratio. 
(Arts.  305,  318.)  In  a  simple  ratio  there  are  but  tico  terms,  an  ante- 
cedent and  a  consequent ;  whereas  in  a  proportion  there  must  at  least 
be  four  terms,  or  two  couplets. 

Again,  one  ratio  may  be  greater  or  less  than  another;  the  ratio  of 
9  to  3  is  greater  than  the  ratio  of  8  to  4,  and  less  than  18  to  2. 
One  proportion,  on  the  other  hand,  cannot  be  greater  or  less  than  an 
other ;  for  equality  does  not  admit  of  degrees. 

QUEST — Obs.  What  are  the  numbers  of  which  a  proportion  is  com- 
posed, called  ?  319.  In  how  many  ways  is  proportion  expressed  1 
What  is  the  first  ?  The  second  I  320.  How  many  terms  must  there 
be  in  a  proportion  ?  Why  ?  Can  a  proportion  be  formed  of  three 
lumbers  ?  How  ?  Will  there  be  four  terms  in  it  ?  Obs.  What  is  the 
number  repeated  cabled  ?  What  is  the  last  term  called  in  such  a  case  t 
What  is  the  difference  between  proportion  and  ratio  ? 


296  PROPORTION.  [SECT. 

321*  The  first  and  last  terms  of  a  proportion  are 
called  the  extremes ;  the  other  two,  the  means. 

OBS.  Homologous  terms  are  either.the  two  antecedents,  or  the  two 
consequents.  Analogous  terms  are  the  antecedent  and  consequent  ol 
the  same  couplet. 

322.  Direct  proportion  is  an  equality  between  two 
direct  ratios.  Thus,  12  :  4  :  :  9  :  3  is  a  direct  propor- 
tion. 

OBS.  In  a  direct  proportion,  the  first  term  has  the  same  ratio  to  the 
second,  as  the  third  has  to  the  fourth. 

323*  Inverse,  or  reciprocal  proportion  is  an  equality 
oetween  a  direct  and  a  reciprocal  ratio.  Thus,  8  :  4  :  :  £  : 
•f ;  or  8  is  to  4,  reciprocally,  as  3  is  to  6. 

OBS.  In  a  reciprocal  or  inverse  proportion,  the  first  term  has  the 
same  ratio  to  the  second,  as  the  fourth  has  to  the  third. 

324*  If  four  numbers  are  proportional,  the  product  oj 
the  extremes  is  equal  to  the  product  of  the  means.  Thus,  8  ; 
4  :  :  6  :  3  is  a  proportion  :  for  f =-|.  (Art.  318.) 

Now      8x3-4x6. 

Again,  12  :  6  :  :  i  :  -J-  is  a  proportion.  (Art.  323.) 

And       12xl=6x|. 

OBS.  1.  The  truth  of  this  proposition  may  also  be  illustrated  in  the 
following  manner : 

The  numbers  2  :  3  :  :  6  :  9  are  obviously  proportional.  (Art.  318.} 
For,  -|=$.  (Art.  120.)    Now, 

Multiplying  each  ratio  by  27,  (the  product  of  the  denominators,) 

2X27^6X27 
The  proportion  becomes     3     :        9      (Art.  284.  Ax.  6.) 


QUEST.— 321.  Which  terms  are  the  extremes  ?  Which  the  means  ? 
Obs.  What  are  homologous  terms  ?  Analogous  terms  I  322.  What  is 
direct  proportion?  Obs.  In  direct  proportion  what  ratio  has  the  first 
term  to  the  second  I  323.  What  is  inverse  proportion  ?  Obs.  What 
ratio  has  the  first  term  to  the  second  in  this  case  ?  324.  If  four  number* 
are.  proportional,  what  is  the  product  of  the  extremes  equal  to  ?  Obs, 
If  the  product  of  the  extremes  is  equal  to  the  product  of  the  means, 
what  is  true  of  the  four  numbers  ?  If  the  products  are  not  equal,  wha* 
is  true  of  the  numbers  ? 


ARTS.  321-326.]  PROPORTION.  297 

Dividing  both  the  numerator  and  the  denominator  of  the  first  coup- 
let by  3;  (Art.  116;)  or  canceling  the  denominator  3  and  the  same 
factor  in  27;  (Art.  136 ;)  also  canceling  the  9,  and  the  same  factor  in 
27,  we  have  2X9—6X3.  But  2  and  9  are  the  extremes  of  the  given 
proportion,  and  3  and  6  are  the  means ;  hence,  the  product  of  the  ex- 
tremes 2X9=6X3,  the  product  of  the  means. 

2.  Conversely,  if  the  product  of  the  extremes  is  equal  to  the  pro- 
duct of  the  means,  the  four  numbers  are  proportional ;  and  if  the  pro- 
ducts are  not  equal,  the  numbers  are  not  proportional. 

325.  Proportion  in   arithmetic,    is   usually  divided 
into  Simple  and  Compound. 

SIMPLE   PROPORTION. 

326.  SIMPLE  PROPORTION  is  an  equality  between  two 
simple  ratios.  (Art.  309.  a.)     It  may  be  either  direct  or 
inverse.  (Arts.  322,  323.) 

The  most  important  application  of  simple  proportion,  is 
the  solution  of  that  class  of  examples  in  which  three  terms 
are  given  to  faid  a  fourth. 

326.  a.  We  have  seen  that,  if  four  numbers  are  in 
proportion,  the  product  of  the  extremes  is  equal  to  the  pro- 
duct of  the  means.  (Art.  324.)  Hence, 

If  the  product  of  the  means  is  divided  by  one  of  the 
extremes,  the  quotient  will  be  the  other  extreme  ;  and  if 
the  product  of  the  extremes  is  divided   by  one  of  the 
means,  the  quotient  will  be  the  other  mean.     For,  if  the 
product  of  two  factors  is  divided  by  one  of  them,  the  quo- 
tient will  be  the  other  factor.  (Art.  291.) 
Take  the  proportion  8  :  4  :  :  6  :  3. 
Now  the  product        8X3-^4=6,  one  of  the  means  ; 
So  the  product  8X3-^6=4,  the  other  mean- 

Again,  the  product     4X6-*-8=3,  one  of  the  extremes ; 
And  the  product         4X6-^-3=8,  the  other  extreme. 


QUEST.- 


325.  Into  what  is  proportion  usually  divided  ?  326.  What 
is  simple  proportion  ?  What  is  the  most  important  application  of  it  ? 
32G.  a.  If  the  product  of  the  means  is  divided  by  one  of  the  extremes, 
what  will  the  quotient  be  ?  If  the  product  of  the  extremes  is  divided  by 
wie  of  the  means,  what  will  the  quotient  be  t 


•298  SIMPLE    §  [SECT  Xll! 

326.  i.  .7/j  therefore,  any  three  terms  of  a  proportion  art 
given,  the  fourth  may  be  found  by  dividing  the  product  of  two 
of  Lhem  by  the  other  term. 

OBS.  Simple  Proportion  is  often  called  the  Rule  of  Three,  from  the 
circumstance  that  three  terms  are  given  to  find  a,  fourth.  In  the 
older  arithmetics,  it  is  also  called  the  Golden  Rule.  But  the  fact  that 
these  names  convey  no  idea  of  the  nature  or  object  of  the  rule,  seems 
to  be  a  strong  objection  to  their  use,  not  to  say  a  sufficient  reason  for 
discarding  them. 

Ex.  1.  If  the  first  three  terms  of  a  proportion  are  4, 
6,  8,  what  is  the  fourth  term  1 

Solution. — 6x8=48  and  48-^4=12,  which  is  the  num- 
ber required ;  that  is,  4  :  6  :  :  8  :  12. 

PROOF.— 4x12  is  equal  to  6x8.  (Art.  324.  Obs.  2.) 

2.  If  12  bbls.  of  flour  cost  $72,  what  will  4  bbls.  cost, 
at  the  same  rate  ? 

Solution. — It  is  evident  12  bbls.  has  the  same  ratio  to  4 
bbls.,  as  the  cost  of  12  bbls.  ($72)  has  to  the  cost  of  4 
bbls.,  which  is  required.  That  is,  12  bbls. :  4  bbls  :  :  $72 
is  to  the  cost  of  4  bbls.  Now,  72x4=288  ;  and  288-*- 12 
=24.  Ans.  $24. 

3.  If  6  men  can  dig  a  cellar  in  12  days,  how  many 
men  will  it  take  to  dig  it  in  4  days  ? 

Note. — Since  the  answer  is  men,  we  put'the  given  number  of  men 
for  the  third  term.  Then,  as  it  will  require  more  men  to  dig  the  cel- 
lar in  4  days  than  it  will  to  dig  it  in  12  days,  we  put  the  larger  num< 
her  of  days  for  the  second  term,  and  the  smaller  for  the  first  term. 

Operation. 

4d.  :  12d.  :  :  6  m.  :  to  the  men  required. 
_6 

4)72 

18  men.  Ans. 


QUEST. — Obs.  What  is  simple  proportion  often  called  ?    Do  theta 
ttenns  convey  on  idea  of  the  nature  or  object  of  the  rule  I 


•I 

£  ART.  327.]  PROPORTION.  299 

327*  From  the  preceding  illustrations  and  principles, 
we  deduce  the  following  general 

RULE  FOR  SIMPLE  PROPORTION. 

I.  Place  that  number  for  the  third  term,  which  is  of  the 
same  kind  as  the  answer  or  number  required. 

IL  Then,  if  by  the  nature  of  the  question  the  answer  must 
be  greater  than  the  third  term,  place  the  greater  of  the  oilier 
two  numbers  for  the  second  term  ;  but  if  it  is  to  be  less,  place 
the  less  of  the  other  two  numbers  for  the  second  term,  and  the 
other  for  the  first. 

III.  Finally,  multiplying  tlie  second  and  third  terms  to- 
gether, divide  the  product  by  the  first,  and  the  quotient  will 
be  the  answer  in  the  same  denomination  as  the  third  term 

PHOOF. — Multiply  the  first  term  and  the  answer  iogetJier, 
and  if  the  product  is  equal  to  the  product  of  the  second  and 
third  terms,  the  work  is  right.  (Art.  324.) 

OBS.  1.  If  the  first  and  second  terms  are  compound  numbers,  re- 
duce them  to  the  lowest  denomination  mentioned  in  either,  before  the 
multiplication  or  division  is  performed.  v 

When  the  third  term  contains  different  denominations,  it  must 
also  be  reduced  to  the  lowest  denomination  mentioned  in  it. 

2.  The  process  of  arranging  the  terms  of  a  question  for  solution, 
that  is,  putting  it  into  the  form  of  a  proportion,  is  called  stating  the 
question. 

3.  We  have  seen  that  questions  in  Simple  Proportion  may  easily 
be  solved  by  Analysis.  (Art.  296.)     After  solving  the  following  exam- 
ples by  proportion,  it  will  be  an  excellent  exercise  for  the  pupil  to 
solve  each  by  analysis. 

4.  If  6  yards  of  broadcloth  cost   30  dollars,  how  much 
will  20  yards  cost?  Ans.  $100. 

5.  If  8  bbls.  of  flour  cost  $40,  what  will  15  bbls.  cost? 

6.  If  16  Ibs.  of  tea  cost  $12,  what  will  41  Ibs.  cost? 

. « 

QUEST. --327.  In  arranging  the  terms  in  simple  proportion,  which  num 
ber  is  put  for  the  third  term  ?  How -arrange  the  other  two  numbers ! 
Having  stated  the  question,  how  is  the  answer  found  ?  Of  what  de- 
nomination is  the  answer  ?  How  is  Simple  Proportion  proved  ?  Obs. 
If  the  first  and  second  terms  contain  different  denominations,  how 
proceed  ?  When  the  third  term  contains  different  denominations,  whaj 
b  to  be  done  ?  What  is  meant  by  stating  tlie  question  ? 


300  SIMPLE  [SECT.  XIL] 

7.  If  12  acres  of  land  produce  240  bushels  of  wheat 
how  much  will  57  acres  produce  ? 

8.  If  a  man  can  travel  400  miles  in  15  days,  how  far 
can  he  travel  in  9  days  ? 

9.  If  63  barrels  of  beef  cost  $504,  how  much  will  7 
barrels  eost  ? 

Common  Method. 

bbls.    bbls.       dolls. 

63  :  7  :  :  504  :  Ans.  Multiplying  the  second  and 

7  third  terms  together  and  divid- 

63)3528($56.  Ans.  in£  the  product  by  the  first, 

315  we  have  $56  for  the  answer. 

"378 
378 

By  Cancelation.  By  canceling  the  factor  7, 

bbls.   bbis.     doiis.  which  is  common  to  the  first 

0$  :  1i  :  :  504  :  Ans.     two  terms  ;  that  is,  which  is 

common  to  the  divisor  and  di- 

Now  504-f-9=$56.  Ans.  vidend,  we  avoid  the  necessity 
of  multiplying  by  it.  (Art.  91.  a.} 

PROOF.—  63x56=504x7.  (Art.  327.)     Hence, 

328.  When  the  first  term  has  one  or  more  factors 
common  to  either  of  the  other  two  terms. 

CANCEL  the  factors  which  are  common,  then  proceed  ac- 
cording to  the  rule  above.  ,(Art.  91.  #.,  136.) 

OBS.  1.  The  question  should  be  stated,  before  attempting  to  cancel 
the  common  factors. 

2.  When  the  terms  are  of  different  denominations,  the  reduction  of 
them  may  sometimes  be  shortened  by  cancelation. 

10.  If  1  2  yds.  of  lace  cost  £  1  ,  what  will  1  qr.  of  a  yard  cost  ? 
Operation. 

y!2x4  :  T  :  .  1x20x12  :  Ans.  (Art.  327.  Obs.  1.) 


Then,  =  =5  d.  Ans. 

12x4 


QUEST.  —  328.  When  the  first  term  has  factors  common  to  either  o! 
the  other  two  terms,  how  may  the  operation  be  shortened  ? 


RT.  328.J  PROPORTION.  301 

*  * 

11.  If  6  men  can  build  a  wall  in  36  days,  how  long 
will  it  take  18  men  to  build  it? 

12.  If  10  quintals  of  fish  cost  $35,  how  much  will  17 
quintals  cost  ? 

13.  If  a  ship  has  water  sufficient  to  last  a  crew  of  25 
men  for  8  months,  how  long  will  it  last  15  men? 

14.  If  1 2  Ibs.  sugar  cast  $  1 ,  how  much  will  84  Ibe.  cost  ? 

15.  If  15  Ibs.  lard  cost  $1. 15,  how  much  will  80  Ibs.  cost? 

16.  Iff  of  an  acre  of  land  cost  ££,  how  much  will  -f 
of  an  acre  cost  ? 


A.      A.      £ 

7 ..  3 

8"7 


"573 
Solution. — _  :  _ : : !  :  to  the  answer. 


Hence,  ?xZx!L=Answer.  (Arts.  327,  139.) 

587 

*        «J         gb  O 

By  cancelation,  (Art.  136,)    LX-X-=—  £-• 

17.  If  -f  of  a  hogshead  of  molasses  cost  $28,  how 
much  will  16  hogsheads  cost? 

18.  If  2i  yds.  of  broadcloth  cost  $18,  how  much  will 
27  yds.  cost  ? 

19.  If  6  acres  and  40  rods  of  land  eost  $125,  how 
much  will  25  acres  and  120  rods  cost? 

20.  If  15  yds.  of  silk  cost  £4,  10s.,  how  much  will  75 
yds.  €ost? 

21.  If  a  Railroad  car  goes  35  m.  in  1  hr.  45  min.,  how- 
far  will  it  go  in  3  days  ? 

22.  If  4£  Ibs.  of  chocolate  cost  9s.,  how  much  will  22-£ 
Ibs.  cost? 

23.  If  35f  Ibs.  of  butter  cost  $4,  how  much  will  15-J 
Ibs.  cost? 

24.  If  84  Ibs.  of  cheese  cost  $5|,  how  much  will  60 
bs.  cost? 

25.  If  f  of  a  ship  is  worth  $6000,  how  much  is  fV  of 
her  worth  ? 

26.  If  4£  bu.  of  wheat  make  1  barrel  of  flour,  how 
many  barrels  will  84  bu.  make  ? 

27.  If  the  interest  of  $1500  for  12  mo.  is  $90,  what 
rill  be  the  interest  of  the  same  sum  for  8  mo.  ? 


• 

30$  COMPOUND  [SECT. 


$8.  If  a  tree  20  ft.  high,  casts  a  shadow  30  ft.  long 
how  long-  will  be  the  shadow  of  a  tree  50  ft.  high  ? 

29.  How  Jong  will  it  take  a  steam  ship  to  sail  round 
the  globe,  allowing  it  to  be  25000  miles  in  circumference, 
if  she  sails  at  the  rate  of  3000  miles  in  12  days  ? 

30.  How  many  acres  of  land  can  a  man  buy  for  $840. 
if  he  pays  at  the  rate  of  $56  for  every  7  acres  ? 

31.  How  much  will  85  cwt.  of  iron  cost,  at  the  rate 
of  $91  for  13  cwt.? 

32.  At  the  rate  of  $45  for  6  cwt.  of  beef,  how  much 
can  be  bought  for  $980  ? 

33.  If  9  ounces  of  silver  will  make  4  tea  spoons,  how 
many  spoons  will  25  pounds  of  silver  make  ? 

34.  If  15  tons  of  wool  are  worth  $90000,  how  much 
is  5  cwt.  worth  ? 

35.  If  5£  yds.  of  cloth  are  worth  $27£,  how  much  are 
50-J-  yards  worth  ? 

36.  If  60  men  can  build   a  house  in  90^-  days,  how 
long  will  it  take  15  men  to  build  it? 

37.  A   bankrupt  owes   $25000,  and  his   property  is 
worth  $20000  :  how  much  can  he  pay  on  a  dollar  ? 

38.  At  7s.  6d.  per  week,  how  long  can  a  man  board 
for  £24,  10s.? 

39.  What  cost  94  tons  of  coal,  if  141  tons  cost  £85? 

40.  What  cost   291  yds.  of  cambric,  if  13    yds.  cost 
£8,  6s.  3£d.  ? 

41.  What  cost  o  Ibs.  of  raisins,  at  £6,  7s.  6d.  per  100  Ibs.  ? 

42.  If  20  sheep  cost  £37,  12-Js.,  what  will  311  cost? 

43.  At  7s.  6d.  per  ounce,  what  is  the  value  of  a  silver 
pitcher  weighing  9  oz.  13  pwt.  8  grs.  ? 

44.  If  405  yards  of  linen  cost  £69,  7s.  6d.,  what  will 
243  yards  cost?' 

45.  If  A  can  saw  a  cord  of  wood  in  6  hours,  and  B  in  9 
hours,  how  long  will  it  take  both  together  to  saw  a  cord  ? 

46.  A  cistern  has  3  cocks,  the  first  of  which  will  empty 
it  in  10  min.  ;  the  second,  in  15  min.  ;  and  the  third,  in 
30  min.  :  how  long  will  it  take  all  of  them  together  to 
empty  it  ? 

47.  A  man  and  a  boy  together  can  mow  an   acre  of 
grass  in  4  hours  ;  the  man  can  mow  it  alone  in  6  hours  : 
how  long  will  it  take  the  boy  to  mow  it  ? 


AitY.  329.]  PROPORTION.  308 

COMPOUND  PROPORTION. 

329.  COMPOUND  PROPORTION  is  an  equality  between 
R.  compound  ratio  and  a  simple  one.     (Arts.  309.  #,  310.) 
Thus, 


Into      6 


„  S  :  :  12  :  3,  is  a  compound  proportion. 


That  is,  8xb  :  4x3  :  :   12  :  3  ;  for,  8x6x3=4x3x12. 

OBS.  Compound  proportion  is  chiefly  applied  to  the  solution  of  ex- 
wnples  which  would  require  two  or  more  statements  in  simple  propor- 
tion. It  is  sometimes  called  Double  Rule  of  Three. 

Ex.  1.  If  4  men  can  earn  $24  in  6  days,  how  much 
can  8  men  earn  in  10  days  ? 

Suggestion. — When  stated  in  the  form  of  a  compound  proportion, 
the  question  will  stand  thus : 

GcT  •  lOd  I  :  :  ®^  :  to  tllc  answer  required.  That  is,  "  the  pro- 
duct of  the  antecedents  4X6,  has  the  same  ratio  to  the  product  of  th« 
consequents,  8X10,  as  $24  has  to  the  answer." 

Operation.  We  divide  the  product  of  all 

24x8x10=1920,  the  numbers  standing  in  the  2d 

and  4x6=24.  and  3d  places  of  the  proportion, 

Now  1920-4-24=80.  by  the  product  of  those  standing 

Ans.  80  dollars.  in  the  first  place. 

Note, — 1.  The  learner  will  observe,  that  it  is  not  the  ratio  of  4  to  8 
alone,  nor  that  of  6  to  10,  which  is  equal  to  the  ratio  of  24  to  the  an- 
swer, as  it  is  sometimes  stated;  but  it  is  the  ratio  compounded  of  4 to 
8  and  6  to  10,  which  is  equal  to  the  ratio  of  24  to  the  answer.  Thus, 
4X6  :  8X10  ::  24  :  80,  the  answer. 

2.  A  compound  proportion,  when  stated  as  above,  is  read,  "  the  ra- 
tio of  4  into  G  is  to  8  into  10  as  24  to  the  answer." 

2.  If  5  men  can  mow  20  acres  of  grass  in  4  days, 
working  10  hours  per  day,  how  much  can  8  men  mow  in 
5  days,  working  12  hours  per  day? 

Operation.  State  the    question,  then 

5m.       8m.     ^    Acwfc  multiply  and  divide  as  be- 

5d.       > : :  20  :  Ans.    fore. 


44. 

lOhr. 


12hr.  \ 


8x5x12x20=9600;  and  5x4x10=200.     Now  9600- 
200=48.  Ans.  48  acres. 


QUEST. — 329.  What  is  compound  proportion  ?     OZ>«.  To  waat  is 
riuefly  applied  ?    What  is  it,  sometimes  called  ? 


304 

33O.  From  the  foregoing  illustrations  we  derive  the 
following  general 

RULE  FOR  COMPOUND  PROPORTION. 

1.  Place  that  number  which  is  of  the  same  kind  as  the  an- 
swer required  for  the  third  term. 

II.  Then  take  the  other  numbers  in  pairs,  or  two  of  a  kind, 
and  arrange  them  as  in  simple  proportion.  (Art.  327.) 

III.  Finally,  multiply  together  all  the  second  and  third 
terms,divide  the  result  by  the  product  of  tite  first  terms,  and  the 
quotient  wUl"be  the  fourth  term  or  answer"  required. 

PROOF. — Multiply  the  ansiver  into  all  of  the  first  terms  or 
antecedents  of  the  first  couplets,  and  if  the  product  is  equal  to 
the  continued  product  of  all  the,  second  and  third  terms  multi- 
plied together,  the  work  is  right.  (Art.  324.) 

OBS.  1.  Among  the  given  numbers  there  is  but  one  which  is  of  the 
*ame  kind  as  the  answer.  This  is  sometimes  called  the  odd  term,  and 
is  always  to  be  placed  for  the  third  term. 

2.  Questions  in  Compound  Proportion  may  be  solved  by  Analysis ; 
also  by  Simple  Proportion,  by  making  two  or  mart  separate  state- 
ments. (Art.  302.  Obs.  327.) 

3.  If  8  men  can  clear  30  acres  of  land  in  63  days, 
working  10  hours  a  day,  how  many  acres  can  10  men 
rlear  in  72  days,  working  12  hours  a  day  ? 


8m. 
63d. 
lOhr. 


'Statement. 

10m.   }       Acrea- 

72d.     >   :  :  30  :  to  the  answer.     That  is, 

12hr. 


8x63X10  :  10X72X12  :  :  30  :  to  the  answer. 
But  the  prod.  10X72X12X30^  Ang    (Art 
Divided  by  8x63x10 


QUEST. — 330.  In  arranging  the  numbers  in  compound  proportion, 
which  number  do  you  put  for  the  third  term  I  How  arrange  the  other 
numbers  ?  HavSig  stated  the  question,  how  is  the  answer  found  ? 
How  are  questions  in  compound  proportion  proved  ?  Obs.  Among  tha 
given  numbers,  how  many  are  of  the  same  kind  as  the  answer  ?  Can 
questions  in  compound  proportion  lx?  solved  by  simple  proportion  ? 
How? 


j 331.]          PROPORTION.  305 

Now  by  canceling  equal  factors,  (Art.  116,)  we  have 
t  Xtf2xl2x30     360 


or  5  If  acres.  Ans.     Hence, 
x*  / 

7 

331*  After  stating  the  question  according  to  the  rule 
above,  if  the  antecedents  or  first  terms  hare  factors  common 
to  the  consequents  or  second  terms,  or  to  the  third  term,  they 
should  be  CANCELED  before  performing  the.  multiplication  and, 
division. 

Note. — Instead  of  placing  points  between  the  first  and  second  terms, 
that  is,  between  the  antecedents  and  consequents  of  the  left  hand 
couplets  of  the  proportion,  it  is  sometimes  more  convenient  to  put  a 
perpendicular  line  between  them,  as  in  division  of  fractions.  (Art.  140.) 
This  will  bring  all  the  terms  whose  product  is  to  be  the  dividend  on  the 
right  of  the  line,  and  those  whose  product  is  to  form  the  divisor,  on 
the  left.  In  this  case  the  third  term  should  be  placed  below  the  se- 
cond terms,  with  the  sign  of  proportion  ( : : )  before  it,  to  show  its 
origin,  and  its  relation  to  the  answer. 

4.  If  a  man  can  walk  192  miles  in  4  days,  traveling 
12  hours  a  day,  how  far  can  he  go  in  24  days,  traveling 
8  hours  a  day? 

Operation.  The  product  of  the  antecedents,  4x12, 

'  '4  d.  2  has  the  same  ratio  to  the  product  of  the 
$  hr.  2  consequents,  24x8,  as  192  has  to  the 
:  192m.  answer  required. 


Ans.  |  192x2x2x2-768  miles. 

5.  If  8  men  can  make  9  rods  of  wall  in  12  days,  how 
many  men  will  it  require  to  make  36  rods  in  4  days  ? 

6.  If  5  men  make  240  pair  of  shoes  in  24  days,  how 
many  men  will  it  require  to  make  300  pair  in  15  days? 

7.  If  60  Ibs.  of  meat  will  supply  8  men  15  days,  how 
orig  will  72  Ibs.  last  24  men  ? 

8.  If  12  men  can  reap   80  acres  of  wheat  in  6  clays, 
how  long-  will  it  take  25  men  to  reap  200  acres? 

9.  If  18  horses  eat  128  bushels  of  oats  in  32  days,  how 
many  bushels  will  12  horses  eat  in  64  days? 

10.  If  8  men  can  build  a  wall  20  ft.  long,  6  ft.  high, 

QUEST. — 331.  When  the  antecedents  have  factors  common  to  the 
toneequenls,  what  should  be  dona  with  them? 


306  DUODECIMALS.  [SECT. 

and  4  ft.  thick,  in  12  days,  how  long-  will  it  take  24  men 
to  build  one  200  ft.  long,  8  ft.  high,  and  6  ft.  thick  ? 

11.  If  8  men  reap  36  acres  in  9  days,  working- 9  hours 
per  day,  how  many  men  will  it  take' to  reap  48  acres  in 
12  days,  working  12  hours  per  day? 

12.  If  $100  gain  $6  in  12  months,  how  long  will  it 
take  $400  to  gain  $18? 

13.  If  $200  gain  $12  in  12  months,  what  will  $400 
gain  in  9  months  ? 

14.  If  8  men  spend  £32  in  13  weeks,  how  much  will 
24  men  spend  in  52  weeks  ? 

15.  If  6  men  can  dig  a  drain  20  rods  long,  6  feet  deep, 
and  4  feet  wide,  in  16  days,  working  9  hours  each  day, 
how  many  days  will  it  take  24  men  to  dig  a  drain  20G 
rods  long,  8  ft.  deep,  and  6  ft.   wide,  working  8  hours 
per  day  ? 


SECTION   XIII. 

DUODECIMALS. 

ART.  332*  DUODECIMALS  are  a  species  of  compound 
numbers,  the  denominations  of  which  increase  and  decrease 
uniformly  in  a  twelvefold  ratio.  Its  denominations  are 
feet,  inches  or  primes,  seconds,  thirds,  fourths,  fifths,  fyc. 

Note. — The  term  duodecimal  is  derived  from  the  Latin  numeral 
duodecim,  which  signifies  twelve. 

TABLE. 

12  fourths      ("")      make  1  third,  marked  '" 

12  thirds  "     1  second,  « 

IS  seconds  "      1  inch  or  prime,     "      in.  or 

12  inches  or  primes       "     I  foot,  "      ft. 

Hence  1'= fV  of  1  foot. 

i"=-Ar  of  i  in.  or  iV  of  iV  of  i  ft.=rlr  of  1  ft. 

\'"—-^  of  1",  or  -1^2  of  iV  of  iV  of  1  ft.=T7W  of  I  ft. 

QUEST. — 332.  What  are  duodecimals  !  What  are  its  denominations 
Note.  What  is  the  meaning  of  the  term  duodecimal  ?   Repeat  the  Table, 
Obs.  What  are  the  accents  called,  which  are  used  to  distinguish  the 
different  denominations  ? 


132-335.]  DUODECIMALS.  307 

OBS.  The  accents  use<l  to  distinguish  the  different  denominations 
below  feet,  are  called  Indices. 

333.  Duodecimals  may  be  added  and  subtracted  in 
the  same  manner  as  other  compound  numbers.    (Arts. 
168,  169.) 

MULTIPLICATION  OF    DUODECIMALS. 

334.  Duodecimals   are  principally   applied   to   the 
measurement  of  surfaces  and  solids.  (Arts.  153,  154.) 

Ex.  1.  How  many  square  feet  are  there  in  a  board  8  ft. 
9  in.  long, and  2  ft.  6  in.  wide? 

Operation.  We  first  multiply  each  denomma- 

8  ft.    9'  length,      tion  of  the  multiplicand  by  the  feet 

2  ft.    6'  width.        in  the  multiplier,   beginning  at   the 

17  ft<    Q'  right  hand.     Thus,  2  times  9'  are  18', 

4  ft.    4'  6"  equal  to  1  ft.  and  6'.     Set  the  6'  un- 

Q1  f    ln,  r>,    A        der  inches,  and  carry  the  1  ft.  to  the 

'    next  product.     2  times  8  ft.  are  16  ft. 

and  1   to  carry  makes  17  ft.     Again,  since  6'=^-  of  a 

ft.  and  9'=A  of  a  ft.,  6'  into  9'  is  -fa  of  a  ft.=54",  or 

4'  and  6".     Write  the  6"  one  place  to  the  right  of  inches, 

and  carry  the  4'  to  the  next  product.     Then  6'  or  -^  of 

a  foot  multiplied  into  8  ft.—  ^  of  a  ft.,  or  48',  and  4'  to 

carry  make  52' ;  but  52'=4  ft.  and  4'.     Now  adding  the 

partial  products,  the  sum  is  21  ft.  10'  61". 

OBS.  It  will  be  seen  from  this  operation,  that  feet  multiplied  into 
feet,  produce  feet;  feet  into  inches,  produce  inches;  inches  into 
inches,  produce  seconds,  &c.  Hence, 

335.  To  find  the  denomination  of  the  product  of  any 
two  factors  in  duodecimals. 

Add  the  indices  of  the  two  factors  together,  and  the  sum 
will  be  the  index  of  their  product. 

Thus,  feet  into  feet,  produce  feet ;  feet  into  inches,  pro- 
duce inches ;  feet  into  seconds,  produce  seconds  ;  feet  into 
thirds,  produce  thirds,  &c. 

QUEST.— 333.  How  are  duodecimos  added  and  subtracted  ?  334.  To 
what  are  duodecimals  chiefly  applied  ?  335.  How  find  the  denomina- 
tion of  the  product  in  duodecim  Is  ?  What  do  feet  into  feet  produce ! 
Feet  into  inches  ?  Feet  into  seconds  ?" 


308  DUODECIMALS.  [SECT. 

Inches  into  inches,  produce  seconds ;  inches  into  seo 
onds,  produce  thirds;  inches  into  fourths,  produce  fifths,  &c. 

Seconds  into  seconds,  produce  fourths  ;  seconds  into 
thirds,  produce  fifths ;  seconds  into  sixths,  piodtrce 
eighths,  &c. 

Thirds  into  thirds,  produce  sixths ;  thirds  into  fifths,  pro 
duce  eighths ;  thirds  into  sevenths,  produce  tenths,  &c. 

Fourths  into  fourths,  produce  eighths ;  fourths  intc 
eighths,  produce  twelfths,  &c. 

Note. — The  foot  is  considered  the  unit,  and  has  no  index. 

336*  From  these  illustrations  we  have  the  following 
RULE  FOR  MULTIPLICATION  OF  DUODECIMALS. 

1.  Place  the  several  terms  of  the  multiplier  under  the  cor- 
responding terms  of  the  multiplicand. 

II.  Multiply  eack  term  of  the  multiplicand  by  each  term 
of  the  multiplier  separately r,  beginning  with  the  lowest  de- 
nomination in  the  multiplicand,  and  the  highest  in  the  mul' 
tiplier,  and  write  the  first  figure  of  each  partial  product  one 
or  more  places  to  tJie  right,  under  its  corresponding  denomi- 
nation. (Art.  335.) 

III.  Finally,  add   the  several  partial  products  togethw, 
carrying  1  for  every  12  both  in  multiplying  and  adding, 
and -the  sum  will  be  the  answer  required. 

OBS.  It  is  sometimes  asked  whether  the  inches  in  duodecimals  are 
linear,  square,  or  cubic.  The  answer  is,  they  are  neither.  An  inch 
is  1  twelfth  of  a  foot.  Hence,  in  measuring  surfaces  an  inch  is  iV 
of  a  square  foot ;  that  is,  a  surface  1  foot  long  and  1  inch  wide.  In 
measuring  solids,  an  inch  denotes  iV  of  a  cubic  foot.  In  measuring 
lumber,  these  inches  are  commonly  called  carpentei*  a  inches. 

2.  How  many  square  feet  are  there  in  a  board  1 8  feet 

9  inches  long,  and  2  feet  6  inches  wide  ? 

3.  How  many  square  feet  are  there  in  a  board  14  feet 

1 0  inches  long,  and  1 1  inches  wide  ? 

QUEST, — What  do  inches  into  inches  produce!  Inches  into  thirds  \ 
Inches  into  fourth*  ?  Seconds  into  seconds  ?  Seconds  into  thirds  ? 
Seconds  into  eighths  ?  Thirds  into  thirds  t  Thirds  into  sixths  ?  336. 
What  is  the  rule  for  multiplication  of  duodecimals  ?  Obs.  What  kind 
of  inches  are  those  spoken  of  in  measuring  surfaces  by  duodecimals  ? 
In  measuring  folids  ?  In  measuring  lumber  what  are  they  called  ? 


LET.  336.]  DUODECIMALS.  309 

4.  How  many  square  feet  in  a  gate  12  feet  5  inches 
tvide,  and  6  feet  8  inches  high? 

5.  How  many  square  feet  in  a  floor  16  feet  6  inches 
.ong,  and  12  feet  9  inches  wide? 

6.  How  many  square  feet  in  a  ceiling  53  feet  6  inches 
long,  and  25  feet  6  inches  wide  ? 

7.  How   many  square  feet  are  there  in  a  stock  of  6 
boards  17  feet  7  inches  long,  and  1  foot  5  inches  wide? 

8.  How  many  feet  in  a  stock  of  10  boards  12  feet  8 
nches  long,  and  1  foot  1  inch  wide? 

9.  How  many  cubic  feet  in  a  stick  of  timber  12  feet 
10  inches  long,  1  foot  7  inches  wide,  and  1  foot  9  inches 
thick  ? 

10.  How  many  cubic  feet  in  a  block  of  marble  8  feet 
4  inches  long,  2  feet  6  inches  wide,  and  1  foot  10  inches 
thick  ? 

11.  How  many  cubic  feet  in  a  load  of  wood  6  feet  7 
inches  long,    3  feet  5  inches  high,  and  3  feet  8  inches 
wide  ? 

12.  How  many  feet  in  a»load  of  wood  7  feet  2  inches 
long,  4  feet  high,  and  3  feet  wide  ? 

13.  How  many  feet  in  a  load  of  wood  9  feet  long,  4 
feet  3  inches  wide,  and  5  feet  6  inches  high  ? 

14.  How  many  feet  in  a  pile  of  wood  100  feet  long, 
5£  feet  high,  and  4  feet  wide  1 

15.  How  many  feet  in  a  pile  of  wood  150  feet  longf, 
8-£  feet  high,  and  5  feet  wide  ? 

16.  How  many  cubic  feet  in  a  wall  40  feet  6  inches 
long,  5  feet  10  inches  high,  and  2  feet  thick? 

17.  How  many  solid  feet  in  a  vat  10  feet  8  inches 
long,  7  feet  2  inches  wide,  and  6  feet  4  inches  deep  ? 

18.  How  many  bricks  8  inches  long,  4  inches  wide, 
and  2  inches  thick,  are  there  in  a  wall  20  feet  long,  10 
feet  high,  and  \\  feet  thick  ? 

19.  How  much  will  the  flooring  of  a  room  which  is  20 
feet  long,   and  18   feet  wide  come  to,  at  6-$-  cents  per 
square  foot  ? 

20.  How  much  will  the  plastering  of  a   wall    16  feet 
square  come  to,  at  12£  cents  per  square  yard  ? 


310 


INVOLUTION. 


[SEC/ 


.  Xff% 


SECTION    XIV. 
INVOLUTION. 

MENTAL      EXERCISES. 

ART.  331.  Ex.  1.  What  is  the  product  of  5  mu.ii 
plied  by  5?  Ans.  5X5=25. 

2.  What  is  the  product  of  3  multiplied  into  3  twice  ? 

Ans.  3X3X3=27. 

3.  What  is  the  product  of  2  into  itself  three  times  ? 

Ans.  2X2X2X2=16. 

338*  When  any  number  or  quantity  is  multiplied 
into  itself,  the  product  is  called  a  power.  Thus,  in  the  ex- 
amples above,  the  products  25,  27,  and  16  are  powers. 

The  original  number,  that  is,  the  number  which  being 
multiplied  into  itself,  produces  a  power,  is  called  the  root 
of  all  the  powers  of  that  number ;  because  they  are  de- 
rived from  it. 

339*  Powers  are  divided  into  different  orders;  as  the 
first,  second,  third,  fourth,  fifth  power,  &c.  They  take  their 
name  from  the  number  of  times  the  given  number  is  used 
QS  Q.  factor,  in  producing  the  given  power. 

Note. — 1.  The  first  power  of  a   number  is  said  to  be  the  number 
itself.     Strictly  speaking,  it  is  not  &  power,  but  a  root.  (Art.  338.) 

3  yards. 

1.  The  second  power  of  a  number  is  also 
called  the  square ;  (Art.  153.  Obs.  1 ;)  for, 
if  the  side  of  a  square  is  3  yards,  then  the  r§ 
product  of  3x3=9  yards,  will  be  the  area  of  ^ 
the  given  square.  (Art.  163.)    But  3X3=9  w 
is  also  the  second  power  of  3 ;  hence,  it  is 
tailed  the  square. 


3X3=9yards. 


QUEST.  —  333.  What  is  a  power  ?  339.  How  are  powers  divided  ? 
From  what  do  they  take  their  name  ?  Note.  What  is  said  to  be  the 
first  power  ?  What  is  the  second  power  called  ?  The  third  1  The 
fourth  ? 


h.  337-340.] 


INVOLUTION. 


3.  The  third  power  of  a  number 
is  also  called   the  cube;   (Art.  154. 
Obs.2;)  for,  if  the  side  of  a  cube  is 
2  feet,  then  the  product  of  2X2X2= 
8  feet,  will  be  the  solidity  of  the  given 
cube.  (Art.  164.)     But  2X2X2=8, 
is  also  the  third  power  of  2;  hence  it 
is  called  the  cube. 

4.  The  fourth  power  of  a  number 
if  culled  the  biquadrate. 


2  feet. 


2X2X2=8  feet. 


4.  What  is  the  square  of  4?  Ans.  16. 

5.  What  is  the  cube  of  3  ?     The  fourth  power  of  3  ? 

6.  The  fourth  power  of  2  ?     The  fifth  power  of  2  ? 

7.  What  is  the  square  of  5  ?     Of  6 1     Of  7  ?     Of  9  ? 
Ot  8?     Of  10?     Of  11?     Of  12? 

8.  What  is  the  cube  of  3  ?     Of  4  ?     Of  5  ?     Of  6  ? 

3  4O.  Powers  are  frequently  denoted  by  a  small  figure 
placed  above  the  given  number  at  the  right  hand. 

This  figure  is  called  the  index  or  exponent.  It  shows 
how  many  times  the  given  number  is  employed  as  a  fac- 
tor to  produce  the  required  power.  Thus, 

The  index  of  the  first  power  is  1,  but  this  is  omitted  ; 
for,  (2)  J=2. 

The  index  of  the  second  power  is  2 ; 

The  index  of  the  third  power  is  3  ; 

The  index  of  the  fourth  power  is  4  ; 

The  index  of  thefiflh  power  is  5  ;  &c.     That  is, 
2*=2,  the  first  power  of  2 ; 
22=2x2,  the  square,  or  2d  power  of  2  ; 
2*  =2x2x2,  the  cube,  or  3d  power  of  2; 
24  =2x2x2x2,  the  biquadrate,  or  4th  power  of  2  ; 
25=2x2x2x2x2,  the  fifth  power  of  2  ; 
2 6  =2x2x2x2x2x2,  the  6th  power  of  2;  &c. 


QUEST. — 3-10.  How  are  powers  denoted  ?  What  is  this  figure  called  ? 
What  does  it  show  ?  What  is  tne  index  of  the  first  power?  Of  tho 
•econd  ?  The  third  ?  Fourth  ?  Fifth  ?  Sixth  ? 


812  INVOLUTION 


EXERCISES   FOR   THE    SLATE. 

9.  Express  the  third  power  of  6  ;  the  4th  power  of  12. 
10.  Express  the  square  of  16 ;  the  cube  of  20  ;  the  fourth 
power  of  25  ;  the  fifth  power  of  72  ;  the  sixth  power  ot 
100  ;  the  tenth  power  of  500. 

341.  The  process  of  finding  a  power  of  a  given 
number  by  multiplying  it  into  itself,  is  called  INVOLUTION 

34:2*  Hence,  to  involve  a  number  to  any  required 
power. 

Multiply  the  given  number  into  itself,  till  it  is  talten  as  a 
factor,  as  many  times  as  there  are  units  in  the  index  of  the, 
power  to  which  the  number  is  to  be  raised.  (Art.  339.) 

OBS.  1.  The  number  of  multiplications  in  raising  a  number  to  any 
given  power,  is  one  less  than  the  index  of  the  required  power. 
Thus,  the  square  of  3  is  written  32,  and  3X3=9,  the  3  is  taken 
twice  as  a  factor,  but  there  is  but  one  multiplication. 

".  A  fraction  is  raised  to  a  power  by  multiplying  it  into  itself. 
Thus,  the  square  of  -|  is  ^•^^==x^i 

Mixed  numbers  should  be  reduced  to  improper  fractions,  or  tlu 
ixOmmon  fraction  may  be  reduced  to  a  decimal. 

3.  All  powers  of  1  are  the  same,  viz:  1;  for  1X1 XI XI,  &c.^l. 

1 1 .  What  is  the  square  of  24  ? 
Common  Operation.  Analytic  Operation. 

24  24=2  tens  or  20+4  units. 

24  24=2  tens  or  20-f4  units. 

96  80+16" 

48  400+80 


576.  Ans.  And  400+160+16=576. 

It  will  be  seen  from  this  operation  that  the  square  o5 
20+4,  contains  the  square  of  the  first  part,  viz :  20x20 
=400,  added  to  twice  the  product  of  the  two  parts,  viz : 
20x4+20x4=160,  added  to  the  square  of  the  last  part, 
viz:  4x4=16.  Hence, 

QUEST. — 341.  What  is  involution  ?  342.  How  is  a  number  involved 
to  any  required  power  ?  Obs.  How  many  rn  duplications  are  there  i* 
raising  a  number  to  a  given  power  ?  How  is  a  fraction  involved  ?  A 
mixed  number  ?  Wha*  are  all  powora  of  1  ? 


ARTS,  jit  1-343.]  EVOLUTION.  313 

^342.  a.   The  square  of  any  number  which  consist?  of 
o  figures,  is  equal  to  the  square  of  the  tens,  added  to  twice 
product  of  the  tens  into  the  units,  added  to  the  square  of 
the  units. 

OBS.  1.  The  product  of  any  two  factors  cannot  have  more  figures 
than  both  factors,  nor  but  one  less  than  both.  For  example,  takej), 
the  greatest  number  which  can  be  expressed  by  one  figure.  (Art.  7.) 
And" (9)2,  or  9X9=81,  has  two  figures,  the  same  number  which 
both  factors  have.  99  is  the  greatest  number  which  can  be  expressed 
by  two  figures;  (Art.  7;)  and  (99)2,  or  99x99=9301,  has  four 
figures,  the  same  as  both  factors  nave. 

Again,  1  i»  the  smallest  number  expressed  by  one  figure,  and  (I)2, 
or  IX  1  =  1)  has  Dut  one  figure  less  than  both  factors.  10  is  the 
smallest  number  which  ran  be  expressed  by  two  figures  ;  and  (10)2, 
or  10X10=100,  has  one  figure  less  than  both  factors.  Hence, 

2.  Any  square  number  cannot  have  more  figures  than  double  the 
number  of  the  root  or  first  power,  nor  but  one  less. 

3.  A  cube  cannot  have  more  figures  than  triple  the  number  of  the 
root  or  first  power,  nor  but  two  less. 

12.  What  is  the  square  of  45?  50?  75?    100?  540? 

13.  What  is  the  cube  of  5  ?     Of  8?   10?   12?  60? 

1 4.  What  is  the  fourth  power  of  3  ?  Of  4  ?  16  ?  20  ? 

15.  What  is  the  fifth  power  of  2?     Of  3  ?    4?    5?  6? 

16.  What  is  the  square  of  i?     Of  i?  -J-?  |?  £?  f? 

1 7.  What  is  the  cube  of  -f  ?     Of  i  ?     Of  $  ?     Of  H  ? 

18.  What  is  the  square  of  2i?     Of3-J-?  5f?   10-f? 

19.  What  is  the  square  of  1.5  ?     Of  3.25  ?     Of  10.25  ? 

EVOLUTION. 

343*  If  we  resolve  25  into  two  equal  factors,  viz: 
5  and  5,  each  of  these  equal  factors  is  called  a  root  of  25. 
So  if  we  resolve  27  into  three  equal  factors,  viz  :  3,  3,  and 
3,  each  factor  is  called  a  root  of  27  ;  if  we  resolve  16  into 
four  equal  factors,  viz :  2,  2,  2,  and  2,  each  factor  is  called 
a  root  of  16.  And,  universally,  when  a  number  is  resolv- 
ed into  any  number  of  equal  factors,  each  of  those  factors 
is  said  to  be  a  root  of  that  number.  Hence, 

QUEST. — 342.  a.  What  is  the  square  of  any  number  consisting  of  two 
figures  equal  to  ?  OBS.  How  many  figures  are  there  in  the  product  of 
any  two  factors  ?  How  many  figures  will  the  square  of  a  number  con- 
tain ?  The  cube  ?  343.  When  a  number  is  resolved  into  any  mim 
ber  of  equal  factors,  what  is  each  of  thoso  factors  called  ? 


314  KVOLIJTiON.  ; 

344:9  A  root  of  a  number  is  a  factor,  which,  beingsS 
multiplied  into  itself  a  certain  number  of  times,  will  pro-  « 
duce  that  number.  (Art.  338.) 

OBS.  When  a  number  is  resolved  into  two  equal  factors,  each  of 
these  factors  is  called  the  second  or  square  root ;  when  resolved  into 
three  equal  factors,  each  of  these  factors  is  called  the  third  or  ?ube 
root ;  when  resolved  into  four  equal  factors,  each  factor  is  called  the 
fourth  root ;  &c.  Hence, 

The  name  of  the  root  expresses  the  number  of  equal  factors  into 
which  the  given  number  is  to  be  resolved. 

For  example,  the  second  or  square  root,  shows  that  the  number  is 
to  be  resolved  into  two  equal  factors ;  the  third  or  cube  root,  into  three 
equal  factors  ;  the  fourth  root,  into  four  equal  factors,  &c.  Thus, 

The  square  root  of  16  is  4;  for  4x4  —  10. 

The  cube  root  of  27  is  3;  for  3X3X3=27. 

The  fourth  root  of  16  is  2;  for  2X2X2X2=16,  &*. 

MENTAL    EXERCISES. 

Ex.  1.  Resolve  25  into  two  equal  factors. 
Solution. — 25=5x5.  A?is.  5,  and  5. 

2.  Resolve  8  into  three  equal  factors. 
Solution.— 8=2x2x2.  Ans.  2,  2,  and  2. 

345*  The  process  of  resolving  numbers  into  equal 
factors  is  called  EVOLUTION,  or  the  Extraction  of  Roots. 

OBS.  1.  Evolution  is  the  opposite  of  involution.  (Art.  341.)  One  is 
finding  a  power  of  a  number  by  multiplying  it  into  itself;  the  other 
is  finding  a  root  by  resolving  a  number  into  equal  factors.  Powers 
and  roots  are  therefore  correlative  terms.  If  one  number  is  a  power 
of  another,  the  latter  is  a  root  of  the  former.  Thus,  27  is  the  cube 
of  3  ;  and  3  is  the  cube  root  of  27. 

2.  The  learner  will  be  careful  to  remember,  that 

In  subtraction,  a  number  is  resolved  into  two  parts; 

In  division,  a  number  is  resolved  into  two  factors; 

In  evolution,  a  number  is  resolved  into  equal  factors. 

3.  What  is  the  square  root  of  16?  Ans.  4. 

4.  What  is  the  square  root  of  36  ?     Of  49  ? 

QUEST. — 344.  What  then  is  a  root  ?  Obs.  What  does  the  name  of 
the  root  express?  What  does  the  square  root  show  ?  The  cube  root? 
The  fourth  root  1  345.  What  is  evolution  ?  Obs.  Of  what  is  it  the 
opposite  ?  Into  what  are  numbers  resolved  in  subtraction  ?  In  divi 
sion  ?  In  evolution  ? 


V 


344-3  46.  J  EVOLUTION.  315 

f5TVV  hat  is  the  square  root  of  64  ?     Of  8  H     Of  1  00  ? 
Of  121?     Of  144? 

6.  What  is  the  cube  or  third  root  of  8  ? 

Soluli&n.  —  If  we  resolve  8  into  three  equal  factors,  each 
of  these  factors  is  2  :  for  2x2x2=8.  The  cube  root  of  8 
therefore,  is  2. 

7..  What  is  the  cube  root  of  27  ? 

8.  What  is  the  cube  root  of  64  1 

9.  What  is  the  cube  root  of  125  ? 
10.  What  is  the  fourth  root  of  16? 
1  1.  What  is  the  square  root  of  -fa  ? 

Solution.  —  The  square  root  of  the  numerator  9,  is  3  ; 
and  the  square  root  of  the  denominator  16,  is  4.  There- 
fore -f-  is  the  square  root  of  -fa  ;  for  -f-  xf  =tV- 

12.  What  is  the  square  root  of  •$•?  Ans.  •§-. 

13.  What  is  the  square  root  of  if  ?     Of  ff? 

14.  What  is  the  square  root  of  f|  ?     Of  -ftfr  ? 

15.  What  is  the  cube  root  of  •£  ?  Ans.  £. 

16.  What  is  the  cube  root  of       l     Of 


346*  Roots  are  expressed  in  two  ways]  one  by  the 
radical  sign  (v)  placed  before  a  number  ;  the  other  by  a 
fractional  index  placed  above  the  number  on  the  right 

hand.     Thus,  v4,or  45  denotes  the  square  or  2d  root  of  4  ; 

3  JL  4  JL 

V27,  or  27  3  denotes  the  cube  or  3d  root  of  27  ;  Vl6,or  164 
denotes  the  4th  root  of  16. 

OBS.  1.  The  figure  placed  over  the  radical  sign,  denotes  the  root,  or 
the  number  of  equal  factors  into  which  the  given  number  is  to  be  re- 
solved. The  figure  for  the  square  root  is  usually  omitted,  and  simply 
the  radical  sign  \/  is  placed  before  the  given  number.  Thus,  the 
square  root  of  25  is  written  \/  25. 

2.  When  a  root  is  expressed  by  ^.fractional  index,  the  denominator 
like  the  figure  over  the  radical  sign,  denotes  the  root  of  the  given  num- 

ber.    Thus.  (25)-  denotes  the  square  root  of  25;  (27)     denotes  the 

cube  root  of  27. 


Quiv-3T. — 316.  In  how  many  ways  are  roots  expressed  ?  What  are 
th«y  ?  Obs.  What  does  the  figure  over  the  radical  sign  denote  ?  What 
f.he  denominator  of  the  fractional  indox  ? 


316  EVOLUTION. 


EXERCISES   FOR   THE    SLATE. 

17.  Express  the  cube  root  of  45  both  ways. 

18.  Express  the  cube  root  of  64  both  ways.     Of  125, 

19.  Express  the  fourth  root  of  181  both  ways.    Of  576 

20.  Express  the  5th  root  of  32  ;  the  6th  root  of  64. 

21.  Express  the  7th  root  of  84 ;  the  8th  root  of  91 ; 
the  9th  root  of  105 ;  the  10th  root  of  256. 

22.  Express  the  cube  root  of  576  ;  the  fourth  root  oi 
675  ;  the  fifth  root  of  1000  ;  the  twelfth  root  of  840. 

347.  A  number  which  can  be  resolved  into  equal 
factors,  or  whose  root  can  be  exactly  extracted,  is  called  a 
perfect  power )  and  its  root  is  called  a  rational  number, 
Thus;  16,  25,  27,  &c.,are  perfect  powers,  and  their  roots 
4,  5,  3,  are  rational  numbers. 

34:8*  A  number  which  cannot  be  resolved  into  equal 
factors,  or  whose  root  cannot  be  exactly  extracted,  is  called 
an  imperfect  power ;  and  its  root  is  called  a  Surd,  or  irra- 
tional number.  Thus,  15,  17,  45,  &c.,  are  imperfect  pow- 
ers, and  their  roots  3.8-f-  ;  4.1-f-;  6.7+,  &c.,  are  surds,  for 
their  roots  cannot  be  exactly  extracted. 

OBS.  A  number  may  be  a  perfect  power  of  one  degree  and  an  im- 
perfect power  of  another  degree.  Thus,  16  is  a  perfect  power  of  the 
second  degree,  but  an  imperfect  power  of  the  third  degree ;  that  is, 
it  is  a  perfect  square  but  not  a  perfect  cube.  Indeed  numbers  are  sel- 
dom perfect  powers  of  more  than  one  degree.  16  is  a  perfect  power 
of  the  3d  and  4th  degrees  :  64  is  a  perfect  power  of  the  2d,  3d  and 
6th  degrees. 

349«  Every  root,  as  well  as  every  power  of  1,  is  1. 
(Art.  342.  Obs.  3.)  Thus,  (1)*,  (I)3,  (I)6,  and  vl,  VJ. 

c 

Vl,  &c.,  are  alt  equni. 


QUEST. — 347.  What  is  a  perfect  power?  What  is  a  rational  num- 
ber ?  348.  What  is  an  imperfect  power  ?  What  is  a  surd  1  Obs.  Ar» 
numbers  ever  perfect  powers  of  one  degree  and  imperfect  pcwers  of 
another  degree  ?  Are  they  often  perfect  powers  of  more  than  one  d« 
gre«  ?  84V.  What  are  all  roots  and  powers  of  1  ? 


Art- 


-350.] 


EVOLUTION. 


317 


jU/  EXTRACTION  OF  THE  SQUARE  ROOT. 

35O.  To  extract  the  square  root ^  is  to  resolve  a  given 
number  into  two  equal  factors ;  or,  to  find  a  number  which 
being  multiplied  into  itself,  will  produce  the  given  number. 
(Art.  344.  Obs.) 

Ex.  1.  What  is  the  side  of  a  square  room  which  con- 
tains 16  square  yards? 

Solution. — Let  the  room  be  re-  4  yards, 

presented  by  the  adjoining  figure. 
It  is  divided  into  1 6  equal  squares, 
which  we  will  call  square  yards. 

Since  the  room  is  square,  the 
question  is  simply  this :  What  is 
the  square  root  of  16?  Now  if 
we  resolve  16  into  two  equal  fac- 
tors, each  of  those  factors  will  be 
the  square  root  of  16.  But  16=4  4x4=  1G  yards. 

X4.     The  square  root  of  16,  therefore,  is  4. 

2.  What  is  the  length  of  one  side  of  a  square  room 
which  contains  576  square  feet  ? 


Operation. 
576(24 
4 

44)176 
176 


Since  we  may  not  see  what  the 
root  of  576  is  at  once,  as  in  the  last 
example,  we  will  separate  it  into  periods 
of  two  figures  each,  by  putting  a  point 
over  the  5,  and  also  over  the  6  ;  that  is, 
over  the  units'  figure  and  over  the  hun- 
dreds. This  shows  us  that  the  root  is  to  have  two  fig- 
ures ;  (Art.  342.  a.  Obs.  2  ;)  and  thus  enables  us  to  find  the 
root  of  part  of  the  number  at  a  time.  Now  the  greatest 
square  of  5,  the  left  hand  period,  is  4,  tl  3  root  of  which 
is  2.  We  place  the  2  on  the  right  hand  of  the  number 
for  the  first  part  of  the  root ;  then  subtract  its  square 
from  5,  the  period  under  consideration,  and  to  the  right  of 
the  remainder  bring  down  76,  the  next  period,  for  a  divi- 
dend. To  find  the  next  figure  in  the  root,  we  double  the  2, 
(he  part  of  the  root  already  found,  and  placing  it  on  the  left 
of  the  dividend  for  a  partial  divisor,  we  find  how  many  times 

What  ft  ft  to  extrart  the  square  root  of  A  mimbw  ? 


S18 


SQUARE  ROOT. 


it  is  contained  in  the  dividend,  omitting  the  right 
ure.  Now  4  is  contained  in  17,  4  times.  Placing  the 
on  the  right  of  the  root,  also  on  the  right  of  the  partial^ 
divisor,  we  multiply  44,  the  divisor  thus  completed,  by  4, 
the  last  figure  in  the  root,  and  subtracting  the  product  176 
from  the  dividend,  find  there  is  no  remainder.  The  an- 
swer therfore  is  24. 

Note.— Since  the  root  is  to  contain  two  figures,  the  2  stands  in  tens' 
place ;  hence  the  first  part  of  the  root  found  is  properly  20 ;  which  be- 
ing doubled,  gives  40  for  the  divisor.  For  convenience  we  omit  the 
cipher  on  the  right;  and  to  compensate  for  this,  we  omit  the  right  hand 
figure  of  the  dividend.  This  is  the  same  as  dividing  both  the  divisor 
and  dividend  by  10,  and  therefore  does  not  alter  the  quotient.  (Art.  88.) 

PFOOF. — 24=2  tens,  or  20+4  units. 
24-2     «          20+4     " 


96 

48 


80- 
400+80 


-16 


(24)2=576     =     400+160+16.   (Art.  342.  a.) 

ILLUSTRATION  BY  GEOMETRICAL  FIGURE. 


20ft. 


II 


Let  the  large  square 
ABCD,  represent  the 
room  in  the  last  exam- 
ple ;  then  the  square  DE 
FG  will  be  the  greatest 
square  of  the  left  hand 
period,  the  root  of  which 
is  20  ft.,  and  20x20= 
400,  the  number  of  feet 
in  its  area.  (Art.  163.) 
But  this  square  400  ft. 
taken  from  576  ft.  leaves 
a  remainder  of  176  ft. 
Now* it  is  plain,  if  this 
remaining  space  is  all  added  to  one  side  of  this  square,  its 
sides  will  become  unequal ;  consequently  it  will  cease 
to  be  a  square.  (Art.  153.  Obs.  1.)  But  if  it  is  equally 
enlarged  on  two  sides  it  will  obviously  continue  to  be  a 

QUEST.— Note.  What  place  does  the  first  figure  of  the  root  occupy  ir» 
the  example  above  ?  Why  is  ths  right  hand  figure  of  the  dividend  omit- 
ted ? 


20ft. 


G 


AET.  3JW  ]  SQUARE  ROOT.  319 

gtBSr  For  this  reason  the  root  is  doubled  for  a  divi- 
fcr  in  the  operation.  The  parallelograms  AEFH  and 
jFIC  will  therefore  represent  the  additions  made  to  the 
two  sides,  each  of  which  is  4  ft.  wide  ;  consequently  the 
area  of  each  is  20x4=80  ft.,  and  the  area  of  both  is 
40x4=  160  ft. 

But  having  made  these  additions  to  two  sides  of  the 
square,  there  is  a  vacancy  at  the  corner.  The  square 
BIFH  represents  this  vacancy,  the  side  of  which  is  4  ft. 
or  the  same  as  the  width  of  the  additions  ;  and  its  area  is 
4x4=16  ft.  For  convenience  of  finding  the  area  of  this 
vacancy,  it  is  customa^  <a  the  operation  to  place  the 
last  figure  of  the  root  on  the  right  of  the  divisor,  and 
thus  it  is  multiplied  into  itself.  The  figure  is  now  a  per  , 
feet  square,  the  length  of  whose  side,  is  20+4=24  ft. 

3  5 1  •  From  these  principles  and  illustrations  we  de 
rive  the  following  general 

RULE  FOR  EXTRACTING  THE  SQUARE  ROOT. 

I.  Separate  the  given  number  into  periods  of  two  figures 
each,  by  placing  a  point  over  the  units1  figure,  a?wiher  over 
the  hundreds,  and  so  on  over  each  alternate  figure. 

II.  Find  the  greatest  square  number  in  tJie  first  or  left 
hand  period,  and  place  its  root  on  the  right  of  the  number 
for  the  first  figure  in  the  root.     Subtract  the  square  of  this 
figure   of  the  root  from   the    period   under  consideration ; 
and  to  the  right  of  the  remainder  bring  down  the  next  peri- 
od for  a  dividend. 

III.  Double  the  root  'just  found  and  place  it  on  the  left 
of  the  dividend  for  a  partial  divisor,  find  hoiv  many  times 
it  is   contained  in  the  dividend ;  omitting  its  right   hand 
figure ;  place  the  quotient   on  the   right   of  the   root,  also 
on  the  right  of  the  partial  divisor ;   multiply  the  divisor 
thus  compkted  by  the  last  figure  of  the  root ;  subtract  the 
product  from   the   dividend,  and   to  the   remainder   bring 
down  the  next  period  for  a  new  dividend  as  before. 

IV.  Double  the  root  already  found  for  a  new  partial  di~ 

QUEST.— 351.  What  is  the  first  step  in  extracting  the  square  root? 
Tte  second «    Third  \    Fourth  ? 


320  SQUARE  ROOT.  [SBC&-XIV 

visor,  diinde^  fyc.  as  before.^  and  thus  continue,  the  o^fanon 
till  the.  root  of  all  the  periods  is  extracted. 

PROOF. — Multiply  the  root  into  itself;  and  if  the  product 
ts  equal  to  the  given  number,  the  work  is  right,  (Art.  344.) 

OBS.  The  product  of  the  divisor  completed  into  the  figure  last  placed 
in  the  root,  cannot  exceed  the  dividend.  Hence,  in  finding  the  figure 
to  be  placed  in  the  root,  some  allowance  must  be  made  for  carrying, 
when  the  product  of  this  figure  into  itself  exceeds  9. 

o  »3  1  •  a.  Demonstration. — The  reason  for  the  several  steps  in 
the  rule  may  easily  be  inferred  from  the  preceding  illustrations.  The 
following  is  a  summary  of  them: 

1.  Separating  the  given  number  into  periods  of  two  figures  each 
shows  how  many  figures  the  root  is  to  contain,  and  thus  enables  us 
to  find  part  of  the  root  at  a  time.  (Art.  342,  a.  Obs.  2.) 

2.  The  square  of  the  first  figure  of  the  root,  is  the  number  effect, 
yards,  &c.  disposed  of  by  the  first  figure  of  the  root ;  it  is  subtracted 
from  the  period  to  find  how  many  feet,  yards,  &c.,  remain  to  be 
added. 

3.  The  root  thus  found  is  doubled  for  a  partial  divisor,  because  the 
addition  must  be  made  on  two  sides  of  the  square  already  found,  or  it 
will  cease  to  be  a  square. 

4.  In  dividing,  the  right  hand  figure  of  the  dividend  is  omitted, 
because  the  cipher  on  the  right  of  the  divisor  is  omitted  ;  otherwise 
the  quotient  would  be  10  times  too  large  for  the  next  figure  in  the  root. 

5.  The  last  figure  of  the  root  is  placed  on  the  right  of  the  divisor 
for  convenience  of  multiplying.     The  divisor  is  then  multiplied  by 
the  last  figure  of  the  root  to  find  the  area  of  the  several  additions  thu» 
made. 


3.  What  is  the  square  root  of  625  ? 

4.  What  is  the  square  root  of  900  ? 

5.  What  is  the  square  root  of  1225  ? 

6.  What  is  the  square  root  of  1 764  1 

7.  What  is  the  square  root  of  2916  ? 

8.  What  is  the  square  root  of  4761  ? 

9.  What  is  the  square  root  of  8649? 
10.  What  is  the  square  root  of  12321  ? 


QUEST. — How  is  the  square  root  proved  ?  Dem.  Why  do  we  separate 
the  given  number  into  periods  of  two  figures  each  ?  Why  subtract  the 
square  of  the  first  figure  in  the  root  from  the  first  period  ?  Why  double 
the  root  thus  found  for  a  divisor  ?  W  hy  omit  the  right  hand  figure  of 
*he  dividend  ?  Why  place  the  last  figure  of  the  root  on  the  right  of  th" 
divisor  ?  Why  multiply  the  divisor  by  the  Inst  figure  in  the  ro<>-  ! 


j?n& 

Air.  i  2:)  SQUARE  ROOT.  321 


hat  is  the  square  root  of  53824  ? 
12.  What  is  the  square  root  of  531441  ? 

352.  If  there  are  decimals  in  the  given  sum,  they 
must  be  separated  into  periods  like  whole  numbers,  by 
placing  a  point  over  units,  then  over  kundredths,  and  so 
on,  over  every  alternate  figure  towards  the  right. 

If  there  is  a  remainder  after  all  the  periods  are  brought 
down,  the  operation  may  be  continued  by  annexing  pe- 
riods of  ciphers. 


OBS.  1 .  There  will  always  be  as  many  decimal  figures  in  the  root, 
as  there  are  periods  of  decimals  in  the  given  number. 

2.  The  square  root  of  a  common  fraction  is  found  by  extracting 
'.he  root  of  the  numerator  and  denominator. 

3.  A  mixed  number  should  be  reduced  to  an  improper  fraction. 
When  either  the  numerator  or  denominator  of  a  common  fraction  is 
not  a. perfect  square,  the  fraction  may  be  reduced  to  a  decimal,  and  the 
approximate  root  be  found  as  above. 


13.  What  is  the  square  root  of  6.25?  Ans.2.5. 

14.  What  is  the  square  root  of  1.96  ? 

15.  What  is  the  square  root  of  29.16  ? 

16.  What  is  the  square  root  of  234.09  ? 

17.  What  is  the  square  root  of  .1225? 

18.  What  is  the  square  root  of  .776161  ? 

19.  What  is  the  square  root  of  2  ? 

20.  What  is  the  square  root  of  \7? 

21.  What  is  the  square  root  of  175  ? 

22.  What  is  the  square  root  of  1 16964  ? 

23.  What  is  the  square  root  of  10316944  ? 

24.  What  is  the  square  root  of 

25.  What  is  the  square  root  of 

26.  What  is  the  square  root  of  6-J-  ? 

27.  What  is  the  square  root  of  52-rV  ? 


QUEST. — 352.  When  there  are  decimals  in  the  given  num^?,  how  are 
they  pointed  off?  When  there  is  a  remainder,  how  proc.?*A  ?  Obs. 
How  do  you  determine  how  many  decimal  figures  there  sho^tl  be  in 
the  root  ?  How  is  the  square  root  of  a  common  fraction  found  ?  G£  a 
mixed  number? 


322  SQUAIIE  ROOT. 

APPLICATIONS   OF  THE   SQUARE 
353*  The  principles  of  the  square  root  may  be  a 
plied  to  the  solution  of  questions  in  which  two  sides 
a  right-angled  triangle  are  given,  and  it  is  required  to  find 
the  other  side. 


354*  A  triangle  is  a  figure 
which  has  three  sides  and  three 
angles,  as  in  the  adjoining  dia- 
gram. 

When  one  of  the  sides  of  a 
triangle  is  perpe?idicular  to  an- 
other side,  the  angle  between 
them  is  called  a  right-a?igk. 
(Legendre,  B.  I.  Def.  12.) 

A  Base.  B 

355.  A  right-angled  triangle  is  a  triangle  which  has 
a  right-angle.  (Leg.  B.  I.  Def.  17.) 

The  side  opposite  the  right-angle  is  called  the  hypoth- 
enuse,  and  the  other  two  sides,  the  base  and  perpendicular. 
The  triangle  ABC  is  right-angled  at  B,  and  the  side  AC 
is  the  hypothenuse. 

3  56.  It  is  an  established  principle  in  geometry,  that 
the  square  described  on  the  hypothenuse  of  a  right-angled 
triangle,  is  equal  to  the  sum  of  the  squares  described  on  the 
other  two  sides.  (Leg.  IV.  1 1 .,  Euc.  I.  47. )  Thus,  if  the  base 
of  the  triangle  ABC,  is  4  feet,  and  the  perpendicular  3 
feet ;  then  the  square  of  4  added  to  the  square  of  3  is  equal 
to  the  square  of  the  hypothenuse  BC  ;  that  is,  (4)2+(3)2, 
or  16+9=25,  the  square  of  the  hypothenuse ;  therefore 
the  square  root  of  25,  which  is  5.  must  be  the  hypothenuse 
itself.  Henee,  when  any  two  sides  of  a  right-angled 
triangle  are  given,  the  third  side  may  be  easily  found. 

QUEST. — 354.  What  i»  a  triangle  ?  What  is  a  right-angle  ?  355. 
What  is  a  right-angled  triangle  ?  Draw  a  right-angled  triangle  upon 
the  black-board.  What  is  the  side  opposite  the  right-angle  called? 
What  are  the  other  two  sides  called  ?  356.  What  is  the  square  de«- 
"*  VKH!  02  «V  hvpothenuse  equal  to  ?  Draw  a  right-angled  triangle,  and 
tatadhe  a  «ma»  cm  «aoh  of  i  IB  sides  t 


Air.  ""59.]         SQUARE  ROOT.  323 

i  &&/!.  When  the  base  and  perpendicular  are  given,  to 
d  the  hypothenuse. 

Add.  the  square  of  the  base  to  the  square  of  the  perpendicular^ 
d'-id  the  square  root  of  the  sum  will  be  the  hypothenuse. 

Thus,  in  the  right-angled  triangle  ABC,  if  the  base  is 
4  and  the  perpendicular  is  3,  then  (4)2-f(3)2=25,  and 
V25=5,  the  hypothenuse. 

358.  When  the  hypothenuse  and  base  are  given,  to 
find  the  perpendicular. 

From  the  square  of  the  hypotlienuse  subtract  the  square  of  the 
base,  and  the  square  root  of  the  remainder  will  be  the  perpen- 
dicular. 

Thus,  if  the  hypothenuse  is  5.  and  the  base  4,  then  (5)* 
— (4)2=^  and  v9=3,  the  perpendicular. 

359.  When  the  hypothenuse  and  the  perpendicular 
are  given,  to  find  the  base. 

From  the  square  of  the  hypothenuse  subtract  the  square  oj 
the  perpendic-iar,  and  the  square  root  of  the  remainder  trill  be 
the  base. 

Thus,  if  the  hypothenuse  is  5,  and  the  perpendicular 
3,  then  (5)2— (3)2=16,  and  Vl6=4,  the  base. 

28.  What  is  the  length  of  a  ladder  which  will  just 
reach  to  the  top  of  a  house  32  feet  high,  when  its  foot  is 
placed  24  feet  from  the  house  ? 

Operation. 

Perpendicular  (32)2=32x32=~1024 
Base  (24)2=24x24=J576^ 

The  square  root  of  their  sum  1600=40.  Ans. 

29.  The  side  of  a  certain  school-room  having  square 
corners,  is  8  yards,  and  its  width  6  yards :  what  is  the 
distance  between  two  of  its  opposite  corners  ? 


QUEST. — 357.  When  the  base  and  perpendicular  are  given,  how  if 
the  hypothenuse  found  I  358.  When  the  hypo'henuse  and  base  are 
given,  how  is  the  perpendicular  found  ?  359.  When  the  hypothemis« 
and  perpendicular  are  given,  h(T\v  is  »h»  bnsefhTJnr? ' 


324  CUBE    ROOT.  XIV 

30.  Two  men  start  from  the  same  place,  one*gS?IH| 
actly  south  40  miles  a  day,  the  other  goes  exactly  west  |fe 
miles  a  day :  how  far  apart  will  they  be  at  the  close  of  tfn^ 
first  day  ? 

31.  How  far  apart  will  the  same  travelers   be  at  the 
end  of  4  days  ? 

32.  A  line  75  feet  long  fastened  to  the  top  of  a  flag  staff 
reaches  the  ground  45  feet  from  its  base :  what  is  the 
height  of  the  flag  staff? 

33.  Suppose  a  house  is  40  feet  wide,  and  the  length  of 
the  rafters  is  32  feet :  what  is  the  distance  from  the  beam 
to  the  ridge  pole  ? 

34.  The  side  of  a  square  field  is  30  rods :  how  far  is  it 
between  its  opposite  corners  ? 

35.  If  a   square  field  contains   10  acres,  what  is  the 
length  of  its  side,  and  how  far  apart  are  its  opposite  corners  I 

EXTRACTION  OF  THE  CUBE  ROOT. 

36O.  To  extract  the  cube  root,  is  to  resolve  a  given  num- 
ber into  three  equal  factors;  or,  to  find  a  number  which  being 
multiplied  into  itself  twice,  will  produce  the  given  number. 
(Art.  345.) 

1.  What  is  the  side  of  a  cubical  block  containing  27 
solid  feet  ? 

Solution. — Let  the  given  block 
be  represented  by  the  adjoining 
cubical  figure,  each  side  of  which  is 
divided  into  9  equal  squares,  which 
we  will  call  square  feet.  Now, 
since  the  length  of  a  side  is  3  feet, 
if  we  multiply  3  into  3  into  3,  the  | 

product  27,  will  be  the  solid  con-   "      

tents  of  the  cube.  (Art.  164.)  "  3x3X3=27- 
Hence,  if  we  reverse  the  process,  i.  e.  if  we  resolve  27 
into  three  equal  factors,  one  of  these  factors  will  be  the 
side  of  the  cube.  (Art.  344.  Obs.)  Ans.  3ft. 

2.  A  man  wishes  to  form  a  cubical  mound  containing 
15625  solid  feet  of  earth  :  what  is  the  length  of  its  side  1 

.  What  is  it  to  extract  the  «uba  root  f 


ARTS.  £60, 361.]  CUBE  ROOT.  325 

nation.  1.  We  first  separate  the  given  num- 

•  ber  into  periods  of  three  figures  each, 

I5b  -D(25  by  p}acing  a  p0int  over  the  units'  figure, 
then  over  thousands.  This  shows  us 
that  the  root  must  have  two  figures, 
(Art.  342.  a.  Obs.  3,)  and  thus  enables 
us  to  find  part  of  it  at  a  time. 

2.  Beginning  with  the  left  hand  pe- 
1525  7625  rioc}?  we  fln(j  the  greatest  cube  of  15  is 
8,  the  root  of  which  is  2.  Placing  the  2  on  the  right  of 
the  given  number  for  the  first  figure  in  the  root,  we  sub- 
tract its  cube  from  the  period,  and  to  the  remainder  bring 
down  the  next  period  for  a  dividend.  This  shows  that  we 
have  7625  solid  feet  to  be  addod  to  the  cubical  mound 
already  found. 

3.  We  square  the  root  already  found,  which  in  reality 
is  20,  for  since  there  is  to  be  another  figure  annexed  to 
it,  the  2  is  tens;  then  multiplying  its  square  400  vby  3, 
we  write  the  product  on  the   left  of  the  dividend  for  a 
divisor ;  and  finding  it  is  contained  in  the  dividend  5  times, 
we  place  the  5  in  the  root. 

4.  We  next  multiply  20,  the  root  already  found,  by 
5,  the  last  figure  placed  in  the  root ;  then  multiply  this 
product  by   3  and  place  it  under  the  divisor.     We  also 
place  the  square  of  5,  the  last  figure  placed  in  the  root, 
under  the  divisor,  and  adding  these  three  results  together, 
multiply  their  sum  1525  by  5,  and  subtract  the  product 
from  the  dividend.     The  answer  is  25. 

PROOF.  (25)3=25x25x25=:"l5625.  (Art.  360.) 
DEMONSTRATION   BY  CUBICAL  BLOCKS. 

361.  The  simplest  method  of  illustrating  the  process  of  ex- 
tracting the  cube  root  to  those  unacquainted  with  algebra  and  geom- 
etry, is  by  means  of  cubical  blocks.* 

1.  Dividing  the  number  into  periods  of  three  figures,  shows   how 

*  A  set  of  these  blocks  contains  1st,  a  cube,  the  side  of  which  is  usually  about 
l|  in.  square ;  2d,  three  side  pieces  about  a  in.  thick,  the  upper  and  lower  bas« 
of  which  is  just  the  size  of  a  side  of  the  cube  ;  3d,  three  corner  pieces,  whose 
ends  are  £  in.  squai-e,  and  whose  length  is  the  same  as  that  of  the  side  pieces : 
4th,  a  small  cube,  the  side  of  which  is  equal  to  the  end  of  the  corner  pieces.  It 
Is  desirable  for  every  teacher  and  pupil  to  have  a  set.  If  not  conveniently  pro* 
fured  at  the  jthops,  any  one  can  easilv  make  them  fin-  himself. 


•326  CUBE  HOOT.  [&x  XIV. 

many  figures  the  root  will  contain,  and  also  enables  us  to  find-.  part  ot 
it  at  a  time.     Now,  placing  the  large  cube  upon  a  table  or  s?rm;i.  -let 
it  represent  the  greatest  cube  in  the  left  hand  period,  whichHrMl 
example  above  is  8,  the  root  of  which  is  2.     We  subtract  this  cul 
from  the  left  hand  period,  and  to  the  remainder  biing  down  the  next 
period,  in  order  to  find  how  many  feet  remain  to  be  added.     In  mak- 
ing this  addition,  it  is  plain  the  cube  must  be  equally  increased  on 
three  sides  ;  otherwise  its  sides  will  become  unequal,  and  it  will  then 
cease  to  be  a  cube.  (Art.  154.  Obs.  2.) 

2.  The  object  of  squaring  the  root  already  found  is  to  find  the  area 
of  one  side  of  this  cube  ;  (Art.  163  ;)  we  then  multiply  its  square  by  3, 
because  the  additions  are  to  be  made  to  three  of  its  sides  ;  and,  divid- 
ing the  dividend  by  this  product  shows  the   thickness  of  these  addi- 
tions.    Now  placing  one  of  the  side  pieces  on  the  top.  and  the  other  two 
on  two  adjacent  sides  of  the  cube,  they  will  represent  these  additions. 

3.  But  we  perceive  there  is  a  vacancy  at  three  corners,  each  of 
which  is  of  the  same  length  as  the  root  already  found,  or  the  side  of 
the  cube,  viz:  20  ft.,  and  the  breadth  and  thickness  of  each  is  5  ft.,  the 
thickness  of  the  side  additions.     Placing  the  corner  pieces  in  these 
vacancies,  they  will  represent  the  additions  necessary  to  fill  them. 
The  object  of  multiplying  the  root  already  found  by  the  figure  last 
placed  in  it,  is  to  obtain  the  area  of  a  side  of  one  of  these  additions  ; 
we  then  multiply  this  area  by  3,  to  find  the  area  of  a  side  of  each  of  them. 

4.  We  find  also  another  vacancy  at  one  corner,   whose   length, 
breadth,  and  thickness  are  each  5  ft.,  the  same  as  the  thickness  of  the 
side  additions.     This  vacancy  therefore  is  cubical.     It  is  represented 
by  the  small  cube,  which  being  placed  in  it,  will  render  the  mound  an 
exact  cube  again.     The  object  of  squaring  5,  the  figure  last  placed  in 
the  root,  is  to  find  the  area  of  a  side  of  this  cubical  vacancy.     We  now 
have  the  area  of  one  side  of  each  of  the  side  additions,  the  area  of  one 
side  of  each  of  the  corner  additions,  and  the  area  of  one  side  of  the  cubical 
vacancy,  the  sum  of  which  is  1525.   Wre  \iext  multiply  the  sum  of  these 
areas  by  the  figure  last  placed  in  the  root,  in  order  to  find  the  cubical  con- 
tents of  the  several  additions.  (Art.  164.)  These  areas  are  added  together, 
and  their  sum  multiplied  by  the  last  figure  placed  in  the  root,  for  the 
sake  of  finding  the  solidity  of  all  the  additions  at  once.     The  result 
would  obviously  be  the  same,  if  we  multiplied  them  separately,  and 
then  subtracted"  the  sum  of  their  products  from  the  dividend. 

3622.  From  the  preceding  illustrations  we  derive  the 
following  general 

QUEST.  —  362.  What  is  the  first  step  in  extracting  the  cube  root  ?  The 
second  ?  Third  ?  Fourth  ?  Fifth  ?  How  is  the  cube  root  proved  ? 
Dem.  Why  separate  the  given  number  into  periods  of  three  figures  <>aeh  ? 


Why  subtract  the  greatest  cube  from  the  left  hand  period  ?  Why  *quare 
tfie  root  already  found  ?  Why  multiply  its  square  by  3  ?  Why  di 
vide  the  dividend  by  this  product?  Why  multiply  the  root  already 

fnnnfl   \\v  tlif»  loct  CifrTnrp  ntapprl    in  it?       "VVhv  mllltinlir  thi<a  r>mr?il^t  Kv  3  ? 


found  by  the  last  figure  placed  in  it  ?  Why  multiply  this  product  by  3* 
Why  square  the  figure  last,  placed  in  the  root  ?  Why  multiply  the  euro 
of  these  areas,  l>y  the  last  figure  placed  in  the  root  ? 


ART.  -  CUBE  ROOT.  327 


,E  FOR  EXTRACTING  THE  CUBE  ROOT. 

I.  Separate  the  give,*,  number  into  periods  of  three  figures 
,  placing  a  point  over  units,  then  over  every  third  fig- 
ure towards  the  left  in  ichole  numbers,  and  over  every  third 
figure  towards  the  right  in  decimals. 

II.  Find  the  greatest  cube  in  t/te  first  period  on  the  lejt 
hand ;  then  placing  its  root  on  the  right  of  the  number  for 
the  first  figure  of  the  root,  subtract  its  cube  from  the  period,  and 
to  the  remainder  bring  down  the  next  period  for  a  dividend. 

III.  Square  the  root  already  found,  regarding  its  local 
value ;  multiply  this  square  by  3,  and  place  the  product  on 
the  left  of  the  dividend  for  a  divisor ;  find  how  many  times 
it  is  contained  in  the  dividend,  and  place  the  result  in  the  root. 

IV.  Multiply  the  root  previously  found,  regarding  its 
local  value,  by  this  last  figure  placed  in  it,  then  multiply 
this  product  by  3,  and  write  the  result  on  the  left  of  the  divi- 
dend under  the  divisor ;    under  this  result  write  also  tht 
square  of  the  last  figure  placed  in  the  root. 

V.  Finally,  add  these  results  to  the  divisor ;  multiply  the 
sum  by  the  last  figure  placed  in  the  root,  and  subtract  the 
product  from  the  dividend.      To  the  right  of  the  remainder 
bring  down  the  next  period  for  a  new  dividend ;  find  a  new 
divisor,  and  proceed  with  the  operation  as  above. 

PROOF. — Multiply  the  root  into  itself  twice,  and  if  the  last 
product  is  equal  to  the  given  number,  the  work  is  right. 

OBS.  1.  When  there  is  a  remainder,  periods  of  ciphers  may  be 
added,  and  the  operation  continued  as  in  square  root. 

2.  If  the  right  hand  period  of  decimals  is  deficient,  this  deficiency 
must  be  supplied  by  ciphers. 

3.  When  there  are  decimals  in  the  given  example,  find  the  root 
as  in  whole  numbers ;  then  point  off  as  many  decimal  figures  in  the 
answer,  as  there  are  periods  of  decimals  in  the  given  number. 

4.  The  cube  root  of  a  common  fraction  is  found  by  extracting  the 
root  of  its  numerator  and  denominator. 

A  mixed  number  should  be  reduced  to  an  improper  fraction. 

5.  W7hen  there  are  more  than  two  periods  in  the  given  example, 
it  is  sufficient  to  annex  mie  cipher  to  the  root  previously  found,  be- 
fore squaring  it  for  the  divisor. 

3.  What  is  the  cube  root  of  1728  ? 


328  EQUATION    OF  [J$$C 


4.  What  is  the  cube  root  of  13824  ? 

5.  If  a  box  in  the  form  of  a  cube, 
solid  inches,  what  is  the  length  of  one  side  ? 

6.  What  is  the  side  of  a  cubical  vat,  which 
57  1787  solid  feet? 

7.  What  is  the  side  of  a  cubical  mound  which  contains 
1953125  solid  yards? 

8.  What  is  the  cube  root  of  2  ? 

9.  What  is  the  cube  root  of  2357947691  ? 
10.  What  is  the  cube  root  of  12.167? 

1  1.  What  is  the  cube  root  of  91.125  ? 

12.  What  is  the  cube  root  of  %\  1 

13.  What  is  the  cube  root  of 


SECTION    XV. 
EQUATION  OF  PAYMENTS. 

ART.  363*  EQUATION  OF  PAYMENTS  is  the  process  of 
finding  the  equalized  or  average  time  when  two  or  more 
payments  due  at  different  times,  may  be  made  at  once, 
without  loss  to  either  party. 

OBS.  The  equalized  or  average  time  for  the  payment  of  several 
debts,  due  at  different  times,  is  often  called  the  mean,  time. 

364.  From  principles  already  explained,  it  is  mani- 
fest, when  the  rate  is  fixed,  the  interest  depends  both  upon 
the  principal  and  the  time.  (Art.  241.)  Thus,  if  a  given 
principal  produces  a  certain  interest  in  a  given  time, 

Double  that  principal  will  produce  tioice  that  interest ; 

HoZf  that  principal  will  produce  half  that  interest ;  &c, 

In  double  that  time  the  same  principal  will  produce 
twice  that  interest ; 

In  half  that  time  the  same  principal  will  produce  ha»J 
that  interest ;  &c. 

QUEST.— 363.  What  is  Equation  of  Payments  ?  Obs.  What  is 
the  average  time  for  the  payment  of  several  debts  sometimes  called  ! 
364.  When  the  rate  is  fixed,  upon  what  does  the  interest  depend  I 


cks66.]  PAYMENTS.  829 

3 

•  Hence,  it  is  evident  that  any  given  principal 
viTl  produce  the  same  interest  in  any  given  time,  as 

^s_jrf/ 

One  half  that  prin.  will  produce  in  double       that  time ; 

One  third  that  prin.  will  "  "  thrice  that  time  ; 
Twice  that  principal  will  "  "  half  that  time  ; 
Thrice  that  principal  will  "  "  a  third  of  that  time,  &c. 

For  example,  at  any  given  per  cent., 

The  int.  of  $2  for  1  year,  is  the  same  as  the  int.  of  SI  for  2  years; 
The  int.  of  $3  for  1  year,  <!  "  "  $1  for  3  years ;  &c. 

The  int.  of  $4  for  1  mo.       "  "  "        ftlfor4mos.; 

The  int.  of  $5  for  1  mo.       "  "        $ 1  for  5  mos ;  &c. 

366.  The  interest,  therefore,  of  any  given  principal  for 
1  year,  or  1  month,  fyc.,  is  the  same,  as  the  interest  of  1  dol- 
lar for  as  many  years,  or  months,  fyc.  as  there  are  dollars 
in  the  given  principal. 

1.  Suppose  you  owe  a  man  $15  and  are  to  pay  him  $5 
in  8  months,  and  $10  in  2  months,  at  what  time  may 
both  payments  be  made  without  loss  to  either  party? 

Analysis. — Since  the  interest  of  $5  for  1  month  is  the 
same  as  the  interest  of  $1  for  5  months,  (Art.  365,)  the 
interest  of  $5  for  8  months  must  be  equal  to  the  interest 
of  $1  for  8  times  5  months.  And  5  mo.  x8=40  mo.  In 
like  manner  the  interest  of  $10  for  1  month  is  equal 
to  the  interest  of  $1  for  10  months,  and  the  interest  of 
$10  for  2  months  is  equal  to  the  interest  of  $1  for  2 
times  10  months.  And  10  mo.  x2=20  months.  Now 
40  months  added  to  20  months  make  60  months  ;  that 
is,  you  are  entitled  to  the  use  of  SI  for  60  months. 
But  $1  is  "fa  of  $15,  consequently  you  are  entitled  to 
the  use  of  $15,  -fa  part  of  60  months,  and  60  months 
-*-15=4.  Ans.  4  months. 

Proof. 

The  interest  of  $5  at  6  per  ct.  for  8  mo.  is  $5  x.04=$.20 
The  interest  of  $10     "      «       "    2  mo.,  is  $10x.01=  .10 

Sum  of  both  $.30 
The  interest  of  $15  at  6  ner  ct.  for  4  mo.  is  15x.02=$.30 


330  EQUATION  OP 

367*  Hence,  we  derive  the  following 

RULE  FOR  EQUATION  OF  PAYMENTS. 

First  multiply  each  debt  by  the  time  before  it  becomes 
due ;  then  divide  the  sum  of  the  products  thus  obtained  by 
the  sum  of  the  debts,  and  the  quotient  will  be  the  average 
time  required. 

OBS.  1.  If  one  of  the  debts  is  to  be  paid  dmcn,  its  product  will  be 
nothing ;  but  in  finding  the  sum  of  the  debts,  this  payment  must  be 
added  in  with  the  others. 

2.  This  rule  is  based  upon  the  supposition  that  discount  and  interest 
paid  in  advance  are  equal.  But  this  is  not  exactly  true;  (Art.  2G1. 
Obs.  1;)  consequently,  the  rule,  though  in  general  use,  is  not  strict- 
ly accurate. 

2.  If  I  owe  a  man  $20,  payable  in  4  months,  $40  pay- 
able in  6  months,  and  $60  in  3  months,  at  what  time 
may  I  justly  pay  the  whole  at  once  ? 

Operation. 

$20x4=$SO,  the  same  as  $1  for  80  mo.  (Art.  366.) 
$40x6-240,     «       "      «  $1  for  240  « 
860x3-180,     «       «      «  $1  for  180  " 

$120debts.500  sum  of  products. 
120)500(4|-  month*  Ans. 

3.  A  merchant  bought  three  lots  of  goods    amounting 
to  $300  ;  for  the  first  he  gave  $100,  payable  in  5  months ; 
for  the  second  $150,  payable  in  8  months ;  for  the  third 
$50,  payable  in  2  months  :  what  is  the  average  time  oi 
all  the  payments  ? 

4.  A  farmer  has  3  notes ;  one  of  $50,  due  in  2  months  ; 
another  of  $100,  due  in  5  months  ;  and  the  third  of  $150, 
due  in  8  months  :  what  is  the  average  time  of  the  whole? 

5.  A  merchant  buys  goods  amounting  to  $1200,  and 
agrees  to  pay  $400  down,  $400  in  4  months,  and  $400 
in  8  months ;  he  finally  concluded  to  give  his  note  for  the 
whole  :  at  what  time  must  the  note  be  made  payable? 

6.  A  man  borrows  $600,  and  agrees  to  pay  $100  in  5 
months,  200  in  5  montns,  and  the  balance  in  8  months* 
when  can  he  justly  pay  the  whole  at  once  ? 

QUEST. — 367.  What  ia  the  rule  for  equation  of  payments  ? 


ARTs/3£7.  368.]  PAYMENTS.  331 


man  buys  a  house  for  $1600,  and  agrees  to  pay 
00  down,  and  the  rest  in  3  equal  annual  instalments: 
hat  is  the  average  credit  for  the  whole  ? 

8.  I  have  $1200  owing  to  me]  $  of  which  is  now  due ; 
£  of  it  will  be  due  in  4  months,  and  the  remainder  in  8 
months  :  what  is  the  average  time  of  the  whole  ? 

9.  A  grocer  bought  goods  amounting  to  $1500,  for 
which  he  was  to  pay  $250  down ;  $300  in  4  mo. ;  and 
$950  in  9  mo. :  when  may  he  pay  the  whole  at  once  ? 

10.  A  young  man  bought  a  farm  for  $2000,  and  agrees 
to  pay  $500  down,  and  the  balance  in  5  equal   annual 
instalments :  what  is  the  average  time  of  the  whole? 

PARTNERSHIP. 

368.  PARTNERSHIP  is  the  associating  of  two  or  more 
individuals  together  for  the  transaction  of  business.  (Art. 
299.)  The  persons  thus  associated  are  called  partners; 
and  the  association  is  termed  a  company  or  firm.  The 
money  employed  is  called  the  capital  or  slock ;  and  the 
profit  or  loss  to  be  shared  among  the  partners,  the  divi/lend. 

CASE    I. 

.Ex.  1.  A  and  B  formed  a  partnership ;  A  furnished 
$300  capital,  and  B  $500  ;  they  gained  $200:  what  was 
each  partner's  share  of  the  gain  ? 

Solution. — Since  the  whole  stock  is  $300-f-$500^$800, 
A  s  part  of  it  was  -|frg-=f ,  and  B's  part  was  i^-=f . 
Now  since  A  put  in  -f  of  the  stock,  he  must  have  -f  of  the 
gain  ;  and  $200x-|=$75.  For  the  same  reason  B  must 
have  1  of  the  gain  ;  and  $200xi=$125. 

PROOF.— $75+ 125=$200,  the  whole  gain.  (Art.  284. 
Ax.  11.)  Hence, 


QUEST.— 368.  What  is  partnership  ?  What  are  the  persons  thus  as- 
eociated  called  ?  What  is  the  association  called  ?  What  is  the  money 
employed  called  ?  What  the  profit  or  loss  ? 


332  PARTNERSHIP,  [SjSf,  XV. 


369*  To  find  each  partner's  share  of  the  ga 
when  the  stock  of  each  is  employed  for  the  same  time. 


Make  each  man's  stock  the  numerator,  and  the  whole  stock 
the  denominator  of  a  common  fraction ;  multiply  the  gain  or 
loss  by  the  fraction  which  expresses  each  man's  share  of  th. 
stock,  and  the  product  will  be  his  share  of  the  gain  or  loss. 

Or,  multiply  each  maiUs  stock  by  the  whole  gain  or  loss  ; 
divide  the  product  by  the  whole  stock,  and  the  quotient  will  be 
his  share  of  the  gain  or  loss. 

PROOF. — Add  the  several  shares  of  the  gain  or  loss  toge- 
ther, and  if  the.  sum  is  equal  to  the  whole  gain  or  loss,  the. 
work  is  right.  (Art.  284.  Ax.  11.) 

OBS.  1.  This  rule  is  applicable  to  questions  in  Bankruptcy,  General 
Average,  and  all  other  operations  in  which  there  is  to  be  a  division 
of  property  in  specified  proportions. 

2.  The  preceding  case  is  often  called  Single  Fellowship.  But  since 
a  partnership  is  always  composed  of  two  or  more  individuals,  it  is 
somewhat  difficult  to  see  the  propriety  of  calling  it  single. 

2.  A,  B,  and  C  entered  into  partnership ;  A  put  in  -fa 
of  the  capital,  B  2\,  and  C  H ;  they  gain  $4800 :  what 
was  each  man's  share  of  the  gain  ? 

3.  A,  B,  and  C  form  a  partnership  ;  A  furnishes  $600, 
B  $800,  and  C  $1000  ;  they  gain  $480  :  what  is  each 
man's  share  of  the  gain  ? 

4.  A  Bankrupt  owes  A  $1200,  B  $2300,  C  $3400,  and 
D  $4500 ;  his  whole  effects  are  worth  $5600  :  how  much 
will  each  creditor  receive  ? 

5.  A,  B,  C,  and  D  make  up  a  purse  to  buy  lottery  tick- 
ets; A  puts  in  $30,  B  $40,  C  $60,  and  D  $70 ;  they 
draw  a  prize  of  $2000 :  what  is  each  man's  share  ? 

6.  A,  B,  and   C  freight  a  vessel  with  a  cargo  worth 
$30000 ;  of  which  A  owned  $8000,  B  $10000,  and  C 
$12000;    in  a   gale  the  master  throws  %  of  the  car gc 
overboard  :  what  was  each  man's  loss  ? 


QUEST. — 369.  How  is  each  man's  share  of  the  gain  or  loss  found, 
when  the  stock  of  each  is  employed  for  the  same  time  ?  How  is  the 
operation  proved  ?  Obs.  To  what  is  this  rule  applicable  ?  What  is  it 
feometirnes  called? 


ARTS.  S^sToTO.j          PARTNERSHIP. 


^^^  CASE   II. 

^.  A  and  B  formed  a  partnership  ;  A  put  m  $300,  and 
**§  $200.     At  the  end  of  2  months  A  took  out  his  stock, 
while  B's  was  employed  6  months;  they  gained  $150: 
tvhat  was  each  man's  just  share  of  the  gain  ? 

Note. — It  is  obvious  that  the  gain  of  each  depends  both  upon  the 
capital  he  furnished,  and  the  time  it  was  employed.  (Art.  364.) 

Solution. — Since  A's  capital  $300,  was  employed  2  mo., 
\iis  share  of  the  gain  is  the  same  as  if  he  had  put  in 
$600  for  1  mo. ;  (Art.  365 ;)  for  $300x2=$600.  Also, 
B's  capital  $200,  being  employed  6  mo.,  his  share  of  the 
gain  is  the  same  as  if  he  had  put  in  $1200  for  1  mo.;  for 
$200x6-$  1200.  The  sum  of  $600  and  $1200  is  $1800. 
A's  share  of  the  gain  must  therefore  be 
B's  "  "  "  "  "  " 
Now  $l50xi=$50,  A's  share. 
And  $150xl=$  100,  B's  share.  Hence, 

3  7  O.  To  find  each  partner's  share  of  the  gain  or  loss, 
when  the  stock  of  each  is  employed  for  different  periods. 

Multiply  each  partner's  stock  by  tJie  time  it  is  employed  ; 
make  each  man's  product  the  numerator,  and  the  sum  of  the 
products  the  d£nominator  of  a  common  fraction ;  then  multiply 
the  whole  gain  or  loss  by  each  man's  fractional  share  of  the 
stock,  and  the  product  will  be  his  share  of  the  gain  or  loss. 

OBS.  This  case  is  often  called  Compound  or  Double  Fellowship. 

8.  A,  B,  and  C  enter  into  business  together  ;  A  puts  in 
$500  for  4  months,  B  $400  for  6  months,  and  C  $800 
for  3  months ;  they  jjain  $340 :  what  is  each  man's  share 
of  the  gain  ? 

9.  A  and  B  hire  a  pasture  together  for  $60  ;  A  put  in 
120  sheep  for  6  months,   and  B  put  in  180  sheep  for  4 
months :  what  should  each  pay  ? 


QUEST. — 370.  When  the  stock  of  each  partner  it  employed  for  dif- 
ferent periods,  how  is  each  man'*  share  found  !  Qb*.  Wliat  ic  thi*  ou* 
Toraotimos  celled ! 


334  EXCHANGE   OP 

10.  The  firm  A,  B,  and  C  lost  $246 ;  A 
$85  for  8  mo.,  B  $250  for  6  mo.,  and  C  $500 
what  is  each  man's  share  of  the  loss  ? 


EXCHANGE   OF  CURRENCIES. 

371*  The  term  currency  signifies  money,  or  the 
Idling  medium  of  trade. 

372.  The  intrinsic  value  of  the  coins  of  different 
nations,  depends  upon  their  weight  and  the  purity  of  the 
metal  of  which  they  are  made.  (Art.  203.  Obs.  1.) 

Note. — 1.  The  present  standard  gold  coins  of  Great  Britain  are  22 
parts  of  pure  gold  and  2  parts  of  copper,  i.  e.  22  carats  fine.  The 
standard  silver  coins  are  37  parts  of  pure  silver  and  3  parts  of  copper. 
A  Pound  Sterling  or  Sovereign  weighs  123.274  grs.,  and  a  shilling, 
(silver,)  3  pwts.  15T3!  grs.* 

2.  For  the  present  standard  weight  and  purity  of  gold  and  silver 
coins  of  the  United  States,  see  Art.  203.  Obs.  2. 

373.  The  relative  value  of  foreign  coins  is  determined 
by  the  laws  of  the  country.     By  act  of  Congress,  1842, 

The  value  of  a  Pound  Sterling,  or  Sovereign.     ....  is  $4.84 

"        "       "    Guinea,  English,  .........  is  5.075 

«        "       "    Franc,  French, .  is  .185 

"        "       "   Five-franc  piece,  (Act  of  18-13,)   ....  is  .93 

«        i,       ,,   Doubloon  of  Spain,  Mexico,   &c.,  of)        ^  lr  535 
standard  weight  and  purity,           $ 

OBS.  1.  The  legal  value  of  a  Pound  Sterling  has  been  changed  sev- 
eral times.  By  the  law  of  1842,  its  value  was  fixed  at  $4.84,  and  it 
now  passes  lor  this  sum  in  all  payments  to  or  from  the  Treasury,  and 
in  reckoning  duties  on  imported  goods  invoiced  in  Sterling  money. 
The  intrinsic  viilue  of  a  £  Sterling  or  sovereign,  is  .$4.801. 

2.  In  1799,  the  value  of  a  Pound  Sterling  was  fixed  at  84.44$ 
which  is  now  called  its  nominal  value. 


QUEST. — 371.  What  is  currency  ?  372.  On  what  does  the  intrinsic 
falue  of  the  coins  of  different  countries  deoend  ?  373.  How  is  the  rela- 
tive value  of  foreign  coins  determined  ?  What  is  the  value  of  a  Pound 
Sterling  ?  Of  a  guinea  ?  A  franc  ?  Five-franc  piec&  ?  A  doubloon  1 

*  Hind's  Arithmetic 


Ar>  "5.]  CURRENCIES.  335 

The  process  of  changing  money  expressed  in 
of  one  country  to  its  equivalent  value 
the  denominations  of  another  country,  is   called    Ex- 
'change  of  Cwre?icies. 

Ex.  1.  Change  £20  sterling  to  Federal  money. 

Suggestion. — Since  £1  is  worth  $4.84,  £20  are  worth 
20  times  as  much  ;  and  $4.84x20=$96.80.  Ans. 

2.  Change  £5,  13s.  6d.  to  Federal  money. 

Operation.  Reduce    13s.    6d, 

£5,  13s.  f3d.=£5.675.    (Art.  200.)     to  the  decimal  of  a 
Value  of  £1—8  4.84  pound,  and  multiply 

Ans.  827.467.  (Art.  215.)      the  sum  bY  $4-84- 

375.  Hence  to  reduce  Sterling  to  Federal  money, 

Set  down  the  pounds  as  whole,  numbers,  and  reduce  ike 
given  shillings,  pence,  and  farthings  to  the  decimal  of  a 
pound;  then  multiply  the  whole  sum  by  $4.84,  (the  value 
of  £1,)  point  off  the  product  as  in  multiplication  of  decimals^ 
mid  it  will  be  the  answer  required. 

OBS.  1.  Guineas,  Francs.  Doubloons,  and  all  foreign  coins,  maybe 
reduced  to  Federal  currency,  by  multiplying  the  given  number  by  the 
value  of  one  expressed  in  Federal  money. 

2.  The  rule  usually  given  for  reducing  Sterling  to  Federal  money, 
is  to  reduce  the  shillings,  pence,  and  farthings  to  the  decimal  of  a 
pound,  and  placing  it  on  the  right  of  the  given  pounds,  divide  tho 
whole  sum  by  -fo.  This  rule  is  based  on  the  law  of  1798,  which  fixed 
the  value  of  a  pound  at  $4.44  y-,  and  that  of  a  dollar  at  4s.  6d.  But 
$4.44i  is  9  per  cent,  of  itself,  or  40  cents,  less  than  $4.84,  which  is 
the  present  legal  value  of  a  pound  ;  consequently,  the  result  or  an- 
swer obtained  by  it,  must  be  9  per  cent,  too  small.  A  dollar  is  now 
»qual  to  49.6d.  very  nearly,  instead  of  54d.  as  formerly. 

3.  What  is  the  value  of  £100  in  Federal  money? 

4.  What  is  the  value  of  £275,  15s.  in  Federal  money  t 

5.  Change  £4507  7s.  6d.  to  Federal  money. 

6.  Change  $27.467  to  Sterling  money. 

Solution. — Since  there  is  £1  in  $4.84,  in  $27.467  there 

QUKST. — 374.  What  is  meant  by  exchange  of  currencies  !  375.  How 
is  Sterling  money  reduced  to  Federal  ?  Obs.  How  may  any  foreign 
e&tas  be  reduced  to  Federal  money  ? 


836  EXCHANGE  OF 

are  as  many  pounds,  as  $4.84  is  contained 

and   $27.467-*-4.84=5.675  ;  that  is,   £5.675. 

the  decimal  .675  to  shillings  and  pence,  (Art.  201,) 

have  £5,  13s.  6d.  for  the  answer.     Hence, 

376*  To  reduce  Federal  to  Sterling  money. 

Divide  the  given  sum  by  $4.84,  (the  value  of  £1,)  and 
point  off"  the  quotient  as  in  division  of  decimals.  The  figures 
on  the  left  hand  of  the  decimal  point  will  be  pounds ;  those 
on  the  right,  decimals  of  a  pound,  which  must  be  reduced  tc 
$hilli?igS)  pence^  and  farthings.  (Art.  201.) 

7.  Change  $486.42  to  Sterling  money. 

8.  Change  $1452  to  Sterling  money. 

376.  a.  In  buying  and  selling  Bills  of  Exchange  on 
England,  the  premium  or  discount  is  commonly  reckoned 
at  a  certain  per  cent,  on  the  nominal  value  of  a  Pound 
Sterling,  which  is  $4.44^  (Art.  373.  Obs.) 

9.  What  is  the  worth  of  a  bill  of  exchange  of  £100  on 
London,  at  9  per  cent,  premium  1 

Solution.— £100x$4.44t=$444.44f,  the  nominal  value, 
Then,  $444.44ix.09=$40.00,  the  premium. 
And  $444.44-f  $40-$484.44.  Ans. 

10.  What  is  the  value  of  £1325,  10s.,  at  8-J-  per  cent, 
premium. 

37  7  •  Previous  to  the  adoption  of  Federal  money  in 
1786,  accounts  in  the  United  States  Avere  kept  in  pounds, 
shillings,  pence,  and  farthings. 

OBS.  At  the  time  Federal  money  was  adopted,  the  colonial  currency, 
or  bills  of  credit  issued  by  the  colonies,  had  more  or  less  depreciated 
in  value :  that  is,  a  colonial  pound  was  worth  less  than  a  pound  Ster- 
ling; a  colonial  shilling,  than  a  shilling  Sterling,  &c.  This  deprecia- 
tion being  greater  in  some  of  the  colonies  than  in  others,  gave  rise  ta 
the  different  State  currencies.  Thus, 

In  New  England  currency,  Va.,  Ky.,  and  Tenn,  6s.  or  £  j^— $1. 
In  New  York  currency,  North  Carolina,  and  Ohio,  8s.  or  £-f-=:$l, 
In  Penn.  cur.,  New  Jer.,  Del.,  and  Md.,  7s.  6d.  (7lb.)  or  £-f =$1, 
In  Georgia  cur.,  and  South  Carolina,  4s.  8d.  (4fs.)  or  £-fa=:'$\< 
In  Canada  currency,  and  Nova  Scotia,  5s.  or  £i=$L 

QUEST. — 376.  How  is  Federal  money  reduced  to  Sterling  ?  377 
jpwvkwa  to  th«  adoption  of  Federal  money,  in  what  w»*a  accounts  kept/ 


ART./S70-379.]  CURRENCIES.  837 


^  deduce  $45  to  New  England  currency. 
lution. — Since  there  are  6s.  in  $1,  in  $45  there  are 
times  6s.     And  6s.x45=270s.     Now  270s.-s-20=-£13, 
10s.  Ans.     Hence, 

378.  To  reduce  Federal  money  to  either  of  the  State 
currencies. 

Multiply  the  given  sum  by  the  number  of  shillings  which, 
in  the  required  currency,  make  $1,  and  the  product  will  be 
the  answer  in  shillings,  and  decimals  of  a  shilling.  The 
shillings  should  be  reduced  to  pounds,  arid  the  decimals  to 
pence  and  farthings.  (Art.  201.) 

12.  Reduce  $378  to  New  England  currency. 

13.  Reduce  $465.45  to  New  York  Cunency. 

14.  Reduce  $640  to  Pennsylvania  currency. 

15.  Reduce  $1000  to  Canada  currency. 

16.  Reduce  £15,    7s.  6d.,  N.  E.  cur.  to  Federal  money. 

Solution. — £15,  7s.6d.=307.5s.(Art.  200.)  Now  since 
bs.  make  $1.  307.5s.  will  make  as  many  dollars,  as  6  is 
contained  times  in  307.5.  And  307.5-^6^851.25.  Ans. 
Hence, 

379.  To  reduce  either  of  the   State  currencies  to 
Federal  money. 

Reduce  the  pounds  to  shillings,  and  the  given  pence  and 
farthings  to  the  decimal  of  a  shilling  ;  then  divide  the  sum 
by  the  number  of  shillings  which,  in  the  given  currency,  make 
81,  and  the  quotient  will  be  the  answer  in  dollars  and  cents. 

17.  Reduce  £48,  15s.,  N.  E.  cur.,  to  Federal  Money. 

18.  Reduce  £73,  4s.,  N.  E.  cur.,  to  Federal  Money. 

19.  Reduce  £100,  18s.,  N.  Y.  cur.,  to  Federal  Money. 

20.  Reduce  £256,  5s.,  N.  Y.  cur.,  to  Federal  Money. 

21.  Reduce  £296,  12s.,  Pcnn.  cur.,  to  Federal  Money. 

22.  Reduce  £430,  8s.,  Penn.  cur.,  to  Federal  Money. 

23.  Reduce  £568,  10s.,  Ga.  cur.,  to  Federal  Money." 

24.  Reduce  £1000, 15s.,  Canada  cur.,  to  Federal  Money. 


QUEST. — 378.  How  is  Federal  Money  reduced  to  the  State  currencies  t 
379.  How  are  the  Krvorol  St*t«  *im0Ticio«  reduced  to  Federal  Money  ? 


338  MENSURATION.  fqfet  XVI. 

SECTION     XVI. 
MENSURATION". 

ART.  3 SO*    MENSURATION  is  the  art    of  measuring 
magnitudes. 

OBS.  The  term  magnitude,  denotes  that  which  has  one  or  more 
of  the  three  dimensions,  length,  breadth,  and  thickness. 

381.  In  measuring  surfaces,  it  is  customary  to  as- 
sume a  square  as  the  measuring  unit,  as  a  square  inch,  a 
square  foot,  a  square  rod,  &c. ;  that  is,  a  square  whose 
side  is  a  linear   unit   of  the    same    name.     (Thomson's 
Legendre,  IV.  4.  Sch.  Art.  153.  Obs.  1.) 

Note. — For  the  demonstration  of  the  following   principles,  seo 
references. 

382.  To  find  the  area  of   a  parallelogram,    and  a 
square.   (Art.  163.  Obs.) 

Multiply  the  length  by  the  breadth.     (Leg.  IV.  5.) 
OBS.  When  the  area  and  one  side  of  a  rectangle  are  given,  the  other 
side  is  found  by  dividing  the  area  by  the  given  side.  (Art.  291.  Note.) 

1.  How  many  acres  are  there  in  a  field  120  rods  long,  and  90 
rods  wide  1  Ans.  07$  acres. 

2.  How  many  acres  in  a  field  800  rods  long,  and  128  rods  wide  7 

3.  Find  the  area  of  a  square  field  whose  sides  are  65  rods  in  length. 

4.  A  man  fenced  off  a  rectangular  field  containing  3750  sq.  rods. 
the  length  of  which  was  75  rods:  what  was  its  breadth  1 

5.  One  side  of  a  rectangular  field  is  1  mile  in  length,  and  the  field 
contains  1GO  acres:  what  is  the  length  of  the  other  side  1 

383.  To  find  the  area  of  a  rhombus.  (Leg.  I.  Def.  18.) 
Multiply  tlie  length  by  the  altitude.  (Leg.  IV.  5.) 
Note. — The  term  altitude,  denotes  perpendicular  height. 

6.  The  length  of  a  rhombus  is  17  ft.,  and  its  perpendicular  height 
12  ft. :  what  is  its  area  1  An&  204  sq.  ft, 

7.  What  is  the  area  of  a  rhombus  whose  altitude  is  25  rods,  aivl 
its  length  28.6  rods'? 

384.  To  find  the  area  of  a  trapezium.    (Leg.  IV.  7.) 
Multiply  half  the  sum  of  the  parallel  sides  by  the  altitude. 

8.  The  parallel  sides  of  a  trapezium  are  15  ft.  and  21  ft.,  and  it» 
altitude  12  ft. :  what  is  its  area  1  Ans.  216  ft. 

9.  Find  the  area  of  a  trapezium  whose  parallel  sides  are  25  rods 
and  37  rods,  and  its  altitude  18  rods. 


ARTS.  j3^j$-3S9.]         MENSURATION.  839 


f.  To  find  the  area  of  a  triangle.  (Leg.  IV.  6.) 

the  base  by  half  the  altitude. 
s.  1.  The  base  of  a  triangle  is  found  by  dividing  the  area  by 
iialf  the  altitude. 

2.  The  altitude  of  a  triangle  is  found  by  dividing  the  area  by  half 
the  base. 

10.  What  is  the  area  of  a  triangle  whose  base  is  45  ft.,  and  its 
altitude  20  ft.?  Ans.  450  sq.  ft. 

11.  What  is  the  area  of  a  triangle  whose  base  is  156  ft.,  and  its 
altitude  63  ft.  1 

386.  To  find  the  area  of  a  triangle,  the  three  sides 
being  given. 

From  half  the  sum  of  the  three  sides  subtract  each  side 
respectively;  then  multiply  together  half  the  sum  and  the 
three  remainders,  and  extract  the  square  root  of  the  product. 

12.  What  is  the  area  of  a  triangle  whose  sides  are  10  ft.,  12  ft., 
and  16  ft.  ?  Ans.  59.92-f-ft. 

13.  What  is  the  area  of  a  triangle  whose  sides  are  each  12  yds.  ? 

387.  To  find  the  circumference  of  a  circle,  when  the 
diameter  is  given.  (Leg.  V.  11.  Sch.) 

Multiply  the  given  diameter  by  3.14159. 

Note. — The  circumference  of  a  circle  is  a  curve  line,  all  the  points 
of  which  are  equally  distant  from  a  point  within,  called  the  centre. 

The  diameter  of  a  circle  is  a  straight  line  which  passes  through 
the  centre,  and  is  terminated  on  both  sides  by  the  circumference. 

The  radius  or  semi-diameter  is  a  straight  line  drawn  from  the 
centre  to  the  circumference. 

14.  What  is  the  circumference  of  a  circle  whose  diameter  is  15  ft.  1 

Ans.  47.123H5  ft. 

15.  What  is  the  circumference  of  a  circle  whose  diameter  is  lOOrods? 

388.  To  find   the  diameter  of  a  circle,   when  the 
circumference  is  given. 

Divide  the  given  circumference  by  3.14159. 

OBS.  The  diameter  of  a  circle  may  also  be  found  by  dividing  the 
area  by  .7vS54,  and  extracting  the  square  root  of  the  quotient. 

10.  What  is  the  diameter  of  a  circle  whose  circumference  is 
34.2477  ft.  1  Ans.  30  ft. 

17.  What  is  the  diameter  of  a  circle  whose  circumference  is 
628.318  yards  1 

389.  To  find  the  area  of  a  circle.  (Leg.  V.  11.) 
Multiply  half  the  circumference  ly  half  the  diameter  • 

or,  multiply  the  circumference  by  a  fourth  of  the  diameter. 


840  MENSURATION.  [ 

Note. — The  area  of  a  circle  may  also  be  found  by  multip 
square  of  its  diameter  by  the  decimal  .7854. 

18.  What  is  the  area  of  a  circle  whose  diameter  is  100  ft.  7 

Ans.  7854  sq.  ft. 

19.  What  is  the  area  of  a  circle  whose  diameter  is  120  rods  ? 

20.  How  many  square  yards  in  a  circle  whose  circumference  is 
160  yards'? 

21.  Required  the  diameter  of  a  circle  containing  50.2656  sq.  rods. 

22.  Required  the  diameter  of  a  circle  containing  201.0624  sq.  ft. 

39O»  The  side  of  a  square  equal  in  area  to  any  given 
surface,  is  found  by  extracting  the  square  root  of  the  given 
surface.  (Arts.  350,  339.  Obs.  2.) 

OBS.  When  it  is  required  to  find  the  dimensions  of  a  rectangular 
field,  equal  in  area  to  a  given  surface,  and  whose  length  is  double, 
triple,  or  quadruple,  &c.,  of  its  breadth,  the  square  root  of  £,  £,  -J, 
of  the  given  surface,  will  be  the  width ;  and  this  being  doubled, 
tripled,  or  quadrupled,  as  the  case  may  be,  will  be  the  kngth. 

23.  What  is  the  side  of  a  square,  whose  area  is  equal  to  that  of  a 
circle  which  contains  225  sq.  yds.  1  Ans.  15  yds. 

24.  What  is  the  side  of  a  square,  whose  area  is  equal  to  that  of  a 
triangle  containing  576  sq.  ft,  1 

25.  The  length  of  a  rectangular  field  containing  80  acres,  is  twice 
its  breadth :  what  are  its  length  and  breadth  1 

39  !•  A.  mean  proportional  between  two  numbers  is 
found  by  multiplying  the  given  numbers  together,  and  ex- 
tracting the  square  root  of  the  product.  (Art.  320.  Obs.  1.) 

26.  What  is  the  mean  proportional  between  9  and  167 

27.  What  is  the  mean  proportional  between  49  and  144  ? 

28.  What  is  the  mean  proportional  between  •*•  and  •*-  ? 

392*  In  measuring  solids,  it  is  customary  to  assume 
a  cube  as  the  measuring  unit,  whose  sides  are  squares  of 
the  same  name.  Thus,  the  sides  of  a  cubic  inch,  are 
square  inches  ;  of  a  cubic  foot,  are  square  feet,  &c.  (Art. 
154.  Obs.  2.) 

OBS.  To  find  the  capacity,  solidity,  or  cubical  contents  of  a  body, 
is  to  find  the  number  of  cubic  inches,  feet,  &c.,  contained  in  the  body 

393*  To  find  the  solidity  of  bodies  whose  sides  are 

perpendicular  to  each  other.  (Art.  164.  Leg.  VII.  11.  Sch.) 

Multiply  the  length,   breadth,   and   thickness   together. 

OBS.  When  the  contents  of  a  solid  body  and  two  of  its  sides  are 
given,  the  other  side  is  found  by  dividing  the  contents  by  the  product 
of  the  two  given  sides.  (Art.  294.) 


A.RTS.   3t)0-397.]  MENSURATION.  341 

2y^1  How  many  cubic  feet  are  there  in  a  stick  of  timber  60  ft.  long, 

wide,  and  "2  ft.  thick  1  Ans.  400  cu.  ft. 

rfSOTHow  many  cubic  feet  in  a  wall  100  ft.  long,  15£  ft.  high,  and 
Pf  ft.  thick? 

31.  A  gentleman  wishes  to  construct  a  cubical  bin,  which  shall 
contain  19683  solid  feet :  what  must  be  the  length  of  its  side  1 

32.  If  a  stick  of  timber  containing  400  cu.  ft.,  is  60  ft.  long,  and 
3£  ft.  thick,  what  is  its  width  1  Ans.  2  ft. 

394.  To  find  the  solidity  of  a  prism. 

Multiply  the  area  of  the  base  by  the  height.  (Leg.  VII.  12.) 
OBS.  1.  This  rule  is  applicable  to  all  prisms,  triangular,  quad- 
rangular, pentagonal,  &c. ;  also  to  all  parallelopipedvns,  whether 
rectangular  or  oblique.  (Leg.  VII.  Def.  4, 8, 9.} 

2.  The  height  of  a  prism  is  the  perpendicular  distance  between 
the  planes  of  the  bases.  Hence,  in  a  right  prism,  the  height  is 
equal  to  the  length  of  one  of  the  sides. 

33.  What  is  the  solidity  of  a  prism  whose  base  is  5  ft.  square,  and 
its  height  15  ft.  1  Ans.  375  cu.  ft. 

34.  What  is  the  solidity  of  a  triangular  prism  whose  height  is 
20  ft.,  and  the  area  of  whose  base  is  460  sq.  ft.  1 

395.  To  find  the  lateral  surface  of  a  right  prism. 
Multiply  the  length  by  the  perimeter  of  the  base. 

OBS.  If  we  add  the  areas  of  both  ends  to  the  lateral  surface,  the 
sum  will  be  the  whole  surface  of  the  prism. 

35.  Required  the  lateral  surface  of  a  triangular  prism  whose  per- 
imeter is  4£  in.,  and  its  length  12  in.  Ans.  54  sq.  in. 

36.  Required  the  lateral  surface  of  a  quadrangular  prism  whose 
sides  are  each  2  ft.,  and  its  length  19  ft. 

396.  To   find   the  solidity   of  a  pyramid,   or  cone. 
(Leg.  VII.  18.  VIII.  4.) 

Multiply  the  area  of  the  base  by  -^  of  the  altitude. 

37.  Required  the  solidity  of  a  square  pyramid,  the  side  of  whose 
base  is  25  ft.,  and  whose  height  is  60  ft.  Ans.  12500  cu.  ft. 

38.  Required  the  solidity  of  a  cone,  the  diameter  of  whose  base 
is  30  ft.,  and  whose  height  is  90  ft. 

397.  To  find  the  lateral  or  convex  surface  of  a  regu- 
lar pyramid,  or  cone.  (Leg.  VII.  16.  VIII.  3.) 

Multiply  the  perimeter  of  the  base  by  %  the  slant-height. 
OBS.  The  slant-height  of  a  regular  pyramid,  is  the  distance  from 
the  vertex  or  summit  to  the  middle  of  one  of  the  sides  of  the  base. 

39.  What  is  the  lateral  surface  of  a  regular  triangular  pyramid 
whose  slant-height  is  10  ft.,  and  whose  sides  are  each  8  ft.  1 

Ans.  120*1.  ft. 


342  MENSURATION.  [3E<.  XVI. 

40.  What  is  the  convex  surface  of  a  cone,  the  perimete 
base  is  500  yds.,  and  whose  slant-height  is  120  }^ds.  1 

398.  To  find  the  solidity  of  a  frustum  of  a  pyrami 
or  cone.  (Leg.  VII.  19.  Sch.,  VIII.  6.) 

To  Ike  sum  of  the  areas  of  the  two  ends,  add  the  square 
root  of  the  product  of  these  areas  j  then  multiply  this  sum 
fy\°f  the  perpendicular  height. 

41.  The  areas  .of  the  two  ends  of  a  frustum  of  a  cone  are  9  sq. 
ft.,  and  4  sq.  ft.,  and  its  height  is  15  ft.  :  what  is  its  solidity  1 

Ans.  95  cu.  ft. 

42.  The  two  ends  of  a  frustum  of  a  pyramid  are  4  ft.  and  3  ft. 
square,  and  its  height  is  10  ft.  :  what  is  its  solidity  1 

399*  The  convex  surface  of  afrustu?n  of  a  pyramid, 
or  cone,  is  found  by  multiplying  half  tlie  sum  of  the  circum- 
ferences of  the  two  ends  by  the  slant-height.  (Leg.  VII.  17.) 

43.  The  circumferences  of  the  two  ends  of  a  frustum  of  a  pyramid 
are  12  ft.  and  8  ft.,  and  its  slant-height  7  ft.  :    what  is  its  convex 
surface  1  Ans.  70  sq.  ft. 

44.  The  circumferences  of  the  two  ends  of  a  frustum  of  a  cone 
are  15  yds.  and  9yds.,  and  its  slant-height,  7  yds.:    what  is  its 
convex  surfaced 

400.  To  find  the  solidity  of  a  cylinder.  (Leg.  VIII.  2.) 
Multiply  tfbe  area  of  the  base  by  the  height  or  length. 

45.  Required  the  solidity  of  a  cylinder  6  ft.  in  diameter,  and  20  ft. 
high.  Ans.  5G5.488cu.il. 

40.  Required  the  solidity  of  a  cylinder  30  ft.  in  diameter,  and 
65  ft.  long. 

401.  To  find  the  convex  surface  of  a  cylinder. 
Multiply  the  circumference  of  the  base  by  the  height. 

47.  What  is  the  convex  surface  of  a  cylinder  16  inches  in  circum 
ference  and  40  in.  long  1  Ans.  610  sq.  in. 

48.  What  is  the  convex  surface  of  a  cylinder,  the  diameter  of 
whose  base  is  20  ft.,  and  whose  height  is  65  ft  1 


To  find  the  surface  of  a  sphere  or  globe. 
Multiply    the    circumference    by  the  diameter.     (Leg. 
VIII.  9.) 

49.  Required  the  surface  of  a  globe  13  inches  in  diameter. 

Ans.  531  sq.  in.  nearly. 

50.  Required  the  surface  of  the  earth,  allowing  its  diameter  to  b«' 
BOOO  miles. 


ARTS,  §98-400.]         MENSURATION.  843 

To  find  the  solidity  of  a  sphere  or  globe. 
the  surface  by  \  of  t/ie  diameter. 


51.  What  is  the  solidity  of  a  globe  12  in.  in  diameter! 

52.  What  is  the  solidity  of  the  earth,  reckoning  its  diameter  at 
8000  miles  1 

4O4»  The  solid  contents  of  similar  bodies  are  to  each 
other,  as  the  cubes  of  their  homologous  sides,  or  like  di- 
mensions. (Leg.  VII.  20.  VIII.  11.  Cor.) 

53.  If  a  ball  4  inches  in  diameter  weighs  32  Ibs.,  what  is  the  weight 
of  a  ball  whose  diameter  is  5  inches  ? 

Solution.—!*  :  53  :  :  32  Ibs.  :  to  the  weight.  Ans.  62.5  Ibs. 

54.  If  a  ball  3  inches  in  diameter  weighs  4  Ibs.,  what  is  the  diam- 
eter of  a  ball  which  weighs  32  Ibs.  1 

4O5»  To  find  the  side  of  a  cube  whose  solidity  shall 
be  double,  triple,  &c.,  that  of  a  cube  whose  side  is  given. 

Cube  the  given  side,  multiply  it  by  the  given  proportion, 
and  the  cube  root  of  the  product  will  be  the  side  of  the  cube 
required. 

55.  What  is  the  side  of  a  cubical  mound,  which  contains  8  times 
as  many  solid  feet  as  one  whose  side  is  3  ft.  Ans.  6  ft. 

56.  Required  the  side  of  a  cubical  vat,  which  contains  16  times 
as  many  solid  feet  as  one  whose  side  is  5  ft. 


GAUGING   OF   CASKS. 

4O6»  To  find  the  contents  or  capacity  of  casks. 

Multiply  the  square  of  the  mean  diameter  into  the 
length  itn,  inches  ;  then  this  -product  multiplied  into  .0034 
will  be  the  wine  gallons  required,  or  multiplied  into  .0028 
will  be  the  beer  gallons. 

OBS.  The  mean  diameter  of  a  cask  is  found  by  adding  to  the  heed 
diameter  .7  of  the  difference  between  the  head  and  bung  diameters 
when  the  staves  are  very  much  curved ;  or  by  adding  .5  when  very 
liUle  curved ;  and  by  adding  .65  when  they  are  of  a  medium  curve. 

57.  How  many  wine  gallons  does  a  cask  contain  whose  length  is 
35  inches,  its  bung  diameter  30  in.,  and  its  head  diameter  26  in., 
it  being  but  little  curved  1  Ans.  93.296  gals. 

58.  How  many  beer  gallons  in  a  cask  54  in.  long,  whose  bung 
diameter  is  42  in.,  and  head  diameter  36  in.,  its  staves  being  much 
curved  1 


344  MISCELLANEOUS    EXAMPLES. 


MISCELLANEOUS  EXAMPLES. 

Ex.  1.  How  much  will  500  sheep  cost,  at  $2£  apiece  1 

2.  How  much  can  a  man  earn  in  240  days,  at  3?i  cts.  per  day  1 

3.  What  will  690  bushels  of  apples  cost,  at  18|  cts.  per  bushel  » 

4.  What  cost  476  cows,  at  $12£  apiece  1 

5.  What  cost  685£  gallons  of  oil,  at  87£  cts.  per  gal.  1 
G.  What  cost  325^  acres  of  land,  at  $10}  per  acre  1 

7.  How  much  flour,  at  $4£  per  bbl.,  can  be  bought  for  $5257 

8.  How  many  yards  of  cloth,  at  $5-^-  per  yard,  can  be  bought  for 
$1230  7  Ans.  240  yds. 

9.  How  many  saddles,  at  $1H,  can  be  bought  for  $5025? 

10.  How  many  horses,  at  $75f ,  can  be  bought  for  $3780  ? 

11.  A  man  bought  -|  of  a  ship,  and  sold  -^  of  it :  how  much  had 
he  left  1  Ans.  -fr. 

12.  A  broker  negotiated  a  bill  of  exchange  of  $10360,  at  1-|  per 
cent. :  what  was  his  commission  1 

13.  What  is  the  interest  of  $2345  for  1  year  and  6  months,  at  G 
per  cent.  1 

14.  What  is  the  int.  of  $1356.25  for  90  days,  at  6  per  ct.  7 

15.  What  is  the  int.  of  $533.11  for  6  months,  at  7  per  ct.  1 

16.  What  is  the  amount  of  $925  for  1  yr.  and  4  mo.,  at  8  perct.7 

17.  What  is  the  amount  of  $4635  for  30  days,  at  7  per  ct.  1 

18.  What  is  the  amount  of  $10360  for  60  days,  at  5  per  ct.  1 

19.  What  is  the  present  worth  of  $1365,  payable  in  6  months, 
when  money  is  worth  7  per  cent,  per  annum  1 

20.  At  6  per  ct.  discount,  what  is  the  present  worth  of  $1623.28, 
due  in  1  year  1 

21.  What  is  the  bank  discount  on  a  note  of  $730,  payable  in  4 
months,  at  6£  per  ct.  1  Ans.  C^'16.212. 

22.  What  is  the  bank  discount  on  a  note  of  $1575,  payable  in  GO 
days,  at  7  per  ct.  1 

23    What  will  35  shares  of  Railroad  stock  cost,  at  10£  per  ct 
advance  7  Ans.  $3867.50. 

24.  What  cost  63  shares  of  bank  stock,  at  3J  per  ct.  discount] 

25.  What  premium  must  a  man  pay  annually  for  insuring  $8500 
on  his  store  and  goods,  at  1^  per  ct.  1 

26.  If  I  obtain  insurance  on  goods,  worth  $16265,  at  2£  per  ct., 
and  the  goods  are  lost,  how  much  shall  I  lose  1 

27.  What  is  the  insurance  on  $925.68,  at  1  %  per  ct.  1 

28.  What  is  the  insurance  on  $63460,  at  -f  per  ct.  1 

29.  What  is  the  insurance  on  $48256,  at  1|  per  ct.  1 

30.  A  man  bought  a  farm  for  $5640,  and  afterwards  sold  it  for  1,1 
per  ct.  more  than  it  cost:  how  much  did  he  make  by  his  bargain  1 

31.  A  merchant  bought  a  stock  of  goods  for  $4390,  and  retailed 
them  at  a  profit  of  22J^  per  ct. :  how  much  did  he  make  1 


MISCELLANEOUS    EXAMPLES.  345 

32.   An  oil  merchant  bought  15000  gallons  of  oil  for  $8500,  and 
sold  it  at  15  per  ct.  advance:  how  did  he  sell  it  per  gal.? 
""Saf  I  buy  1675  yards  of  flannel  for  $368.50,  how  must  I  retail 
r  yard  to  gain  25  per  ct.  7  Ans.  27£  cts. 

4.  'A  grocer  bought  2500  Ibs.  of  coffee  for  $250,  and  sold  it  at  b 
per  ct.  loss :  what  did  he  get  per  pound  7 

35.  A  merchant  bought  1824  yds.  of  cloth,  at  $2.50  per  yd.,  and 
retailed  it  at  $3  per  yd. :  what  per  ct.  was  his  profit,  and  how  much 
did  he  make  7 

36.  A  shop-keeper  bought  100  pieces  of  lace,  for  $250,  and  sold 
them  for  $375 :  what  per  ct.  did  he  make  7 

37.  If  a  grocer  buys  3680  Ibs  of  cheese,  at  4£  cts.  per  lb.,  and 
sells  it  at  6^  cents,  what  per  ct.  is  his  profit  7 

38.  What  is  the  ad  valorem  duty,  at  33^  per  ct.,  on  a  quantity  of 
cloths  which  cost  $104367 

39.  W7hat  is  the  ad  valorem  duty,  at  15 £  per  ct.,  on  a  cargo  of 
tea  invoiced  at  $35856  7 

40.  At  37^  per  ct.,  what  is  the  duty  on  a  quantity  of  silks  which 
cost  $23265  7 

41.  The  sum  of  two  numbers  is  856,  and  their  difference  is  75: 
what  are  the  numbers  7 

42.  The  sum  of  two  numbers  is  5643,  and  their  difference  is  125: 
what  are  the  numbers  7 

43.  The  difference  of  two  numbers  is  63,  and  the  smaller  number 
is  365  :  what  is  the  greater  number  7 

44.  The  product  of  two  numbers  is  3750,  and  one  of  the  numbers 
is  75  :  what  is  the  other  7 

45.  What  number  is  that  -f-  of  which  is  265  7  Ans.  477. 

46.  What  number  is  that  -f  of  -f  of  which  is  120  7 

47.  How  long  will  it  take  a  person  to  count  a  billion,  if  he  counts 
50  a  minute,  and  works  6  hours  per  day,  for  5  days  a  week,  and  52 
weeks  a  year  7 

48.  How  many  dollars,  each  weighing  412^  grains,  can  be  made 
from  7  Ibs.  1  oz.  18  pwt.  I8grs.  of  silver  7 

49.  How  many  pound^pf  silk  will  it  take  to  spin  a  thread  which 
will  reach  round  the  earth,  allowing  its  circumference  to  be  25000 
miles,  and  2^  oz.  to  make  160  rods  of  thread? 

50.  How  many  times  will  the  hind  wheel  of  a  carriage,  7  ft.  6  in. 
in  circumference,  turn  round  in  7  miles,  1  furlong,  30  rods? 

51.  How  many  times  will  the  fore  wheel  of  a  carriage,  5  ft.  7£  in. 
in  circumference,  turn  round  in  the  same  distance  7 

52.  What  cost  645  bushels  of  salt,  at  4s.  N.  Y.  currency  per  bu.  7 

53.  What  cost  744  yards  of  muslin,  at  Is.  4d.  N.  Y.  cur.  per  yd.  7 

54.  What  cost  241  melons,  at  2s.  8d.  N.  Y.  cur.  apiece  7 

55.  What  cost  1536  yards  of  calico,  at  Is.  N.  E.  cur.  per  yd.  ? 

56.  What  cost  873  baskets  of  peaches,  at  3s.  N.  E.  cur.  a  basket? 

57.  What  cost  632  bushels  of  oats,  at  Is.  6d.  N.  E.cur.  a  bushel? 

58.  What  cost  848  lambs,  at  5s.  sterling  apiece  ? 

59.  What  cost  258  yards  of  cloth,  at  15s.  sterling  per  yard? 


346         '  MISCELLANEOUS    EXAMPLES. 

60.  What  cost  912  bushels  of  rye,  at  2s.  Gd.  sterling  r_  _ 

61.  What  cost  657  yards  of  silk,  at  (is.  8d.  ster.  per  yard? 

62.  What  co^t  735  bushels  of  apples,  at  Is.  8d.  ster   per  busTTefCt^ 

63.  What  cost  3  pieces  of  cloth,  each  containing  27  yards,  oHj 
3s.  4d.  per  yard?  Aits.  £13,  10s.    * 

64.  What  cost  248  pair  of  boots,  at  12s.  6d.  sterling  a  pair? 

65.  If  156  Ibs.  of  butter  cost  $15.60,  what  will  730  Ibs.  cost? 

66.  If  48  yards  of  cloth  cost  $480,  what  will  125  yards  cost  ? 

67.  If  96  horses  eat  192  tons  of  hay  in  a  winter,  how  many  ton* 
will  ISOJiorses  eat? 

6*8.  If  10  Ibs.  of  sugar  cost  9fs.,  what  will  240  Ibs.  cost? 
69.  If  25  Ibs.  of  veal  cost  $|,  how  much  will  872  Ibs.  cost? 
70    If  50  Ibs.  of  ginger  cost  $7f,  how  much  will  460  Ibs.  cost? 

71.  What  cost  260  cords  of  wood,  if  45  cords  cost  $87f  ? 

72.  A  man  sold  a  sheep  for  £l^,  and  a  pig  for  -|s.  -|d. :  what  did 
he  get  for  both  ? 

73.  A  goldsmith  melted  up  $  Ib.  10^  pwts.  of  gold,  at  one  time,  and 
3£  oz.  lOgrs.  at  another:  how  much  did  he  melt  in  all  ? 

74.  A  man  having  2$  oz.  of  silver,  sold  6|  pwts. :    how  much 
had  he  left? 

75.  A  man  owing  £| ,  2-^-s.,  paid  7-J-s.  2^-d. :  how  much  does  ha 
still  owe  ? 

76.  If  50  Ibs.  of  rice  cost  £*-,  what  will  840  Ibs.  cost  ? 

77.  If  13  yards  of  edging  cost  $^-9.,  what  will  200  yds.  cost  ? 

78.  If  -f-  of  a  ton  of"  iron  cost  $35,  what  will  381  tons  cost? 

79.  If  I  owe  a  man  £6950,  and  can  pay  him  but  13s.  4d.  on  a 
pound,  how  much  will  he  receive  for  his  debt  ? 

80.  If  385  yards  of  linen  cost  £63,  how  much  can  be  bought 
for  £18? 

81.  How  much  brondy  can  be  bought  for  £396,  if  90  gallons 
cost  £18? 

82.  If  15^  yards  silk  cost  $18|,  what  will  56|  yards  cost  ? 

83.  A  grocer  used  a  false  weight  of  13£  oz.  for  a  pound:  what 
was  the  amount  of  his  fraud  in  weighing  500  pounds  ? 

84.  If  f-  of  a  barrel  of  apples  costs  $4 ,  how  much  will  -£  of  a  bar- 
rel cost  ?  Ans.  $2.45. 

85.  If  -jZg  of  a  pound  of  lard  costs  -[•*•  of  a  shilling,  how  much 
will  |-2-  af  a  pound  cost  ? 

86.  If  -fa  of  a  ton  of  hay  costs  £-£,  what  will  -J-g-  of  a  ton  cost  ? 

87.  How  much  will  -fa  of  a  drum  of  figs  come  to,  at  the  rate  of 
f-  of  a  dollar  for  •£•  of  a  drum  ? 

88.  Bought  48  £  Ibs.  of  tea  for  $27| :  how  much  can  be  bought 
for  SI  25? 

89.  Paid  $35^-  for  -£•  of  an  acre  of  land :  how  much  can  be  bought 
for  $7500? 


• 

MISCELLANEOUS    EXAMPLES.  347 

50.  i&Svit  yards  of  camlet  make  3  cloaks,  how  many  cloaks  can 

of  7:>74  yards  1  A/is.  75  cloaks. 

If  57.35  acres  of  land  produce  430.16  bushels  of  barley,  how 
ny  bushels  will  172.05  acres  produce  7 

92.  What  will  730f  yards  of  cloth  cost,  if  you  pay  $112  for 
yards '] 

93.  If  a  cane  3  feet  in  length  cast  a  shadow  5  feet  long,  how  high 
Is  a  steeple  whose  shadow  is  175  feet  7 

91.  Bought  a  hogshead  of  molasses  for  4  firkins  of  butter,  each 
containing  06  Ibs.,  which  was  worth  10  cents  a  pound :  what  did 
the  molasses  cost  per  gallon  ] 

95.  Bought  15  yds.  of  silk  at  7s.  per  yard,  and  12  yds.  of  muslin 
at  3s.  per  yard,  and  paid  the  bill  in  cheese  at  9d.  per  pound :  how 
many  pounds  did  it  take  to  pay  the  bill  ? 

96.  If  a  cubic  foot  of  pure  water  weighs  1000  oz.,  what  will  a  pail 
of  water  weigh  which  contains  217£  cubic  inches'? 

97.  If  I  pay  $8400  for  |  of  a  ship,  what  must  I  pay  for  the 
whole  ship  1 

98.  A  farmer  sold  174  sheep,  which  was  -|-  of  all  he  had;  the 
remainder  he  divided  equally  between  his  two  sons :  how  many  did 
each  receive  1 

99.  A  garrison  having  been  besieged  108  days,  found  that  -f  of 
the  provisions  were  consumed  :  how  much  longer  would  they  last  1 

100.  A  garrison  of  1520  men  have  416955  Ibs.  of  flour:  how  long 
will  it  last  them,  allowing  each  man  -f-  Ib.  per  day  1 

101.  How  long  will  75240  gals,  of  water  last  a  ship's  company 
of  30  men,  allowing  each  man  -^  gal.  per  day  1 

102.  If  10  men  can  dig  a  cellar  in  30  days,  how  long  will  it  take 
25  men  to  dig  it  1 

103.  If  6  men  spend  $48  in  7  weeks,  how  much  will  24  men 
spend  in  35  weeks  1  Ans.  $960. 

104.  If  15  horses  consume  70  bushels  of  oats  in  27  days,  how 
many  bushels  will  45  horses  consume  in  54  days'? 

105.  If  6  men  can  build  a  wall  30  feet  long,  G  feet  high,  and  3 
feet  thick,  in  15  days,  when  the  days  are  12  hours  long,  how  many 
days  will  it  take  30  men  to  build  a  wall  300  feet  long,  8  feet  high, 
and  6  feet  thick,  working  8  hours  a  day  1 

106.  A  merchant  in  New  York  wished  to  pay  £1500 in  London: 
what  will  a  bill  of  exchange  cost  him  at  9  per  ct.  premium  1 

107.  A  broker  in  Boston  sold  a  bill  of  exchange  on  Liverpool  for 
£2500,  15s.,  at  9£  per  ct.  premium:  what  did  he  get  for  it1? 

108.  What  will  a  bill  on  England  for  £3125,  12s.  6d.  cost,  when 
exchange  is  10  per  ct.  above  par  1 

109.  A  man  wishing  to  remit  $2550  to  Ireland,  bought  a  draft  on 
London,  at  12£  per  ct.  advance :  what  was  the  amount  of  his  bill 
in  sterling  money  1 

110.  A  farmer  wishes  to  form  a  square  field,  which  shall  contain 
1296  squ've  rods :  what  is  the  length  of  its  side? 


348  MISCELLANEOUS    EXAMPLES. 

111.  A  man  owns  a  farm  which  contains  160  acres,  and  is  in  the 
form  of  a  square :  what  is  the  length  of  its  side  1 

112.  What  is  the  length  of  the  side  of  a  square  field  containing 
10  acres  ?  ,  f 

113.  What  is  the  area  of  a  triangle  whose  hypothenuss  is  ife 
yards,  and  its  perpendicular  30  yards  '? 

114.  What  is  the  area  of  a  triangle  whose  hypothenuse  is  100 
tods,  and  its  base  60  rods  7 

1 15.  Required  the  mean  proportional  between  49  and  81. 

116.  Required  the  mean  proportional  between  121  and  5.76. 

1 17.  What  is  the  mean  proportional  between  •£  and  -J-f  1 

1 18.  Required  the  mean  proportional  between  -f|-  and  T8^. 

119.  A  regiment  containing  6912  soldiers,  was  so  arranged  that 
the  number  in  rank  was  triple  that  in  file  :  how  many  were  there 
in  each  1 

120.  If  a  board  is  8  in.  wide,  how  long  must  it  be  to  make  a  sq.  ft  1 

121.  How  much  silk  f  yd.  wide  will  it  take  to  make  a  sq.  yd.  1 

122.  How  much  cambric  £  yd.  wide  will  it  take  to  line  9  yds.  of 
balzorine  1  yd,  wide? 

123.  How  many  yds.  of  unbleached  muslin  f  yd.  wide  will  it  take 
to  line  36  yds.  of  carpeting  \\  yds.  wide  1 

124.  If  it  takes  10  yds.  of  broadcloth  1£  yds.  wide  to  make  a  cloak, 
how  many  yards  of  camlet  f  yd.  wide  will  make  one? 

125.  How  much  will  it  cost  to  carpet  a  parlor  18  ft.  square  with 
carpeting  f  yd.  wide,  which  is  worth  $1.50  per  yard  1 

126.  A,  B,  and  C,  joined  in  a  speculation;   A  put  in  $500,  B 
$700,  and  C  put  in  the  balance :  they  gained  $1200,  of  which  C 
received  $480  for  his  share :  how  much  did  A  and  B  receive,  and 
how  much  did  C  put  in  1 

127.  A,  B,  and  C,  gain  $3600,  of  which  A  receives  $6,  as  often 
as  B  receives  $10,  and  C  $14:  what  was  the  share  of  each? 

128.  The  hour  and  minute  hand  of  a  clock  are  exactly  together 
&t  noon  :  when  will  they  next  be  together  ? 

129.  A  farmer  having  lost  ^  of  his  sheep,  and  sold  ^  of  them,  had 
500  left:  how  many  had  he  at  first  ? 

130.  If  -£•  of  a  post  stands  in  the  mud,  \  in  the  .vater,  and  10  feet 
above  the  water,  what  is  the  length  of  the  post  1 

131.  Two  persons  start  from  the  same  place,  one  goes  south  4 
miles  per  hour,  the  other  west  5  miles  per  hour :  how  far  apart  are 
they  in  9  hours  ? 

132.  A  messenger  traveling  8  miles  an  hour,  was  sent  to  Mexico 
with  dispatches  for  the  army ;  after  he  had  gone  51  miles,  another 
was  sent  with  countermanding  orders,  who  could  go  19  miles  at 
quick  as  the  former  could  go  16:  how  long  will  it  take  the  latter  to 
overtake  the  former ;  and  how  far  must  he  travel  1 


ANSWERS    TO    EXAMPLES. 


NOTE. — At  the  urgent  request  of  several  distinguished  Teach- 
ers, who  have  received  Thomson's  Practical  Arithmetic  with 
favor,  the  publishers  have  issued  an  edition  of  it,  containing  the 
answers  in  the  end  of  the  book.  It  is  hoped  that  pupils,  who  may 
use  this  edition,  will  have  sufficient  regard  to  their  own  improve- 
ment, never  to  consult  the  answer  till  they  have  made  a  strenu- 
ous and  persevering  effort  to  solve  the  problem  themselves. 

N.  B. — The  work  without  the  answers  is  published  as  here- 
tofore. 


ADDITION. 

EXERCISES   FOR   THE    SLATE. — ART.    21. 


Ex.   ANS. 

Ex.   ANS. 

Ex.    Am. 

Ex.   ANS. 

1,  2.  Given. 

21.  16840. 

15.  582  a. 

34.  $512. 

3.  8786. 

22.  220083. 

16.  $45. 

Q,   }  611  bu. 

4.  8689. 

23.  100003. 

17.  98  cts. 

d0'  ?  $513. 

fi.  57757. 

24.  134735. 

18.  $101. 

36.  $627. 

6.  651465. 

25.  104022. 

19.  $2788. 

37.  630  Ibs. 

7.  8651761. 

20.  $102. 

38.  $3789. 

8.  998943483 

ART.  29. 

21.  $846. 

39.  $1125. 

9.  988. 

1.  $64. 

C  754  sh. 

40.  $2385  r. 

10.  7673. 

2.  46  Ibs. 

22.  I  365  1. 

$554  g. 

11.  88765. 

3.  48  yrs. 

(  1119  b. 

41.  $1582. 

12.  85879944. 

4.  $313. 

23.  $6821. 

42.  $1323. 

13-15.  Given. 

5.  $31. 

24.  $2324. 

43.  525  m. 

6.  40  s. 

25.  $4900. 

44.  $4930. 

ART.  27. 

7.  $550. 

26.  $244. 

45.  2234822. 

16.  23770. 

8.  $2480. 

27.  113  ts. 

46.  4604345. 

17.  161524. 

9.  $190. 

28.  476  m. 

47.  5067843. 

18.  131570. 

10.  $278. 

29.  73  yrs. 

48.  4984097. 

19.  1999990. 

11.  $58. 

30.  $1648. 

49.  178346. 

12.  33  sch. 

31.  $34950. 

50.  17069453, 

ART.  2§. 

13.  $136. 

32.  $33700. 

30.  1913. 

14.  64m. 

33.  $3147. 

350 


A  N  S  W  E  R  S . 


5.,  34-54, 


SUBTRACTION. 


Ex.        ANS. 

Ex.        ANS. 

Ex.        ANS. 

Ex.        An«. 

ART.  34. 

20.  6121. 

13.  $1291 

33.  $5250. 

1,  2.  Given. 

21.  2754087. 

14.  53  m. 

34.  $323. 

3.  $232. 

22.  932417. 

15.  93  m. 

35.   1933  a. 

4.  413. 

23.  6834501. 

jg_  

36.  565  men. 

5.  353. 

24.  8960895. 

17.   1706. 

37.  $773. 

6.  418. 

25.  31090814 

18.  67  yrs. 

38.  $18053. 

7.  3332. 
8.  3231. 

ART.  4O. 

19.  

20.  $72320. 

39.  $154. 
40.  $5491. 

9.  32352. 

1.  13  yds. 

21.  427721. 

41.  $6749. 

10.  613134. 

2.  $221. 

22.  214412. 

42.  $1695. 

11.  531141. 

3.  189  g. 

23.   1056109. 

43.  $2752. 

12.  3151721. 

4.  1003  bu. 

24.  194099. 

44.  $1913. 

13-15.  Given. 

5.  ,$3791. 

25.  11763528. 

45.  $332. 

ART.  38. 
16.  54182. 
17.   124907. 
18.  66104149. 

6.  $1420. 
7.  $382. 
8.  $1079. 
9.  $374  bu. 
10.  $1989. 

26.  100  a. 
27.  $986. 
28.  $22. 
29.  $19. 
30.   146  ts. 

46.  12520  bu. 
47.  $1491. 
48.  $9699. 
49.  $21422. 
50.  $8000. 

ART.  39. 

11.  $479. 

31.  $1090. 

19.  Given. 

12.  

32.  $3838. 

MULTIPLICATION. 


ART.  47. 

16.  5200  s. 

ART.  54. 

15.  $2522. 

1  A       r^-i  \rnri 

17.  40030. 

16.  $2090. 

1  "1  •     VTlVvll* 

6QfiO   r 

18.  608240. 

1.  $2790. 

17.  4935  s. 

•    you  r  * 

6    880  m. 

19.  76342. 

2.  $2552. 

18.  3071  bu. 

7     QflQfi 

20.  41479110. 

3.  $9520. 

19.  2944  qts. 

/.    yvjt/o. 

SOQAQf) 

21,  22.  Given. 

4.  676  s. 

20.  $22224. 

•    OO'ioU* 

9*0^05 

5.  2511  s. 

21.  $1482. 

•     tlUt/tlUi/. 

10.  9036906. 

nf-rl  VATI 

ART.  53. 

6.  $13932. 
7.  $10955. 

22.  $8991. 
23.  $10584. 

•       VJflVtsil* 

8.  $3790. 

24.  $4096. 

ATTT     *^1 

23.  2268  s. 

9.  $153900. 

25.  35720  d. 

/YKr.    OJL« 

24.  3915  bu. 

10.  $180. 

26.  16425  d. 

12.  $664. 

25.  19200  Ibs. 

11.  $414. 

27.  90625  Ibs. 

13.  1917  s. 

26.  $6394. 

12.  $945. 

28.  176175  lb^ 

14.  $624. 
15.  $6153. 

27-29.  Given.   |13.  $1792. 
30.  507166416.114.  $1664. 

29.  78475  m. 
30.  $77970. 

Ams. 


ANSWERS  . 


$51 


JONTRACTIONS  IN  MULTIPLICATION. 
ARTS.  55-61. 


%£\.    ANS. 

Ex.       ANS. 

Ex.       ANS. 

1.  Given. 

13.  476000. 

26.  390677500000. 

2.  $2295. 

14.  534860000. 

27.  Given. 

3.  Given. 

15.  1204670800000. 

28.  11840000. 

4.  $684. 

16.  26900785000000. 

29.  373520000. 

5.  $4950. 

17.  890634570000000. 

30.  3603200000. 

6.  1872  s. 

18.  946030506800000. 

31.  55447000000. 

7.  8610  m. 

19.  783120650730000. 

32,  33.  Given. 

8.  25760  bu. 

20,  21.  Given. 

34.  4059360000. 

9.  16128  s. 

22.  1080  d. 

35.  14760000000. 

10.  $91080. 

23.  38400  Ibs. 

36.  6204000000. 

11.  Given. 

24.  10940000. 

37.  16726^0000000. 

12.  ?5200  p. 

25.  2075994000. 

38.  1075635900000. 

SHORT  DIVISION.— ARTS.  67-73. 


1,  2.  Given  . 

14.  12212. 

26.  8111. 

39.  71000. 

3.  7. 

15.  11111. 

27.  911. 

40.  Given. 

4.  6. 

16.  1243143. 

28,  29.  Given. 

41.  $107. 

5.  6. 

17.  Given. 

30.  48  Ibs. 

42.  2050. 

6.  9. 

18.  31. 

31.  7615. 

43.  5070. 

7.  Given. 

19.  61. 

32.  6573. 

44.  5021. 

8.  123  sh. 

20.  51. 

33.  16334. 

45.  80405. 

9.  124  a. 

21.  312. 

34.  3144. 

46,  47.  Given. 

10.  122  tms. 

22.  8231. 

35.  107  bbls. 

48.  151£. 

11.  Given. 

23.  711. 

36.  6010. 

49.  52  y. 

12.  321  yds. 

24.  7111. 

37.  7000. 

50.  162|. 

13.  21312. 

25.  811. 

38.  5100. 

LONG  DIVISION.—  ARTS.  74-76. 

1-4.  Given. 

18.  Given. 

8.  11  t. 

20.  588,  &  8  r. 

5.  127208£. 

19.  1080-£f. 

9.  11  c. 

21.  24,&61r. 

6.  1342314. 

20.  901-Hf. 

10.  120  m. 

22.  8,  &  13  r. 

7.  326561. 

11.  200  m. 

23.  227,  &  5  r. 

8.  336568. 

ART.  77. 

12.  250  m. 

24.  269,  &  1  r. 

9.  6437612. 

1.  24  h. 

13.  13^2  mos. 

25.  2813,  &  9  r. 

10.  72225723. 

2.  36  yds. 

14.  20  hhcls. 

26.  34,  &  34  r. 

11-13.  Given. 

3.  43  c. 

15.  11  m. 

27.  173,&25r. 

14.  245. 

4.  108t 

16.  14,  &  4  r. 

28.  158,&40r. 

15.  1326f-§-. 

5.  13  m. 

17.  42. 

29.  388,&55r. 

16.  1212f|-. 

6  20  d. 

18.  39,  &  7  r. 

30.  63,  &  72  r. 

17,  123744- 

7.  lO^rrn, 

19.  72,  &  12  r., 

352 


ANSWERS. 


124. 


CONTRACTIONS  IN  DIVISION. 


Ex.        ANS. 

Ex.                  ANS. 

Ex.            ANS.  'v4 

ARTS.  T8-81. 
1,  2.  Given. 
3.  6  p. 
4.  35  c. 
5.  7-A-. 
6.  3f£. 

7.  Given. 
8.  10  ;  25  ;  38  d. 
9.  65  ;  765  ;  $4320. 
10.  Given. 
11.  44,  &  360791  r. 
12.  8236,  &  7180309  r. 

13.  Given. 
14.  8  h. 
15.  34  bbls. 
16.  210  r. 
17.  68-rV-Llu1-. 

21,  22.  Given. 

23.  76. 

24.  75. 


CANCELLATION.— ART.  91. 


25,26.  Given. 

27.  475. 

28.  798. 


29.  1248. 

30,  31.  Given. 
32.  27. 


33.  7. 

34.  28. 

35.  30. 


GREATEST  COMMON  DIVISOR.— ARTS.  94-97. 


1.  Given. 

2.  3. 

3.  4. 


4.  5. 

5.  4. 

6.  3. 


7.  Given. 

8.  21. 

9.  13. 


10.  19. 

11.  15. 

12.  39. 


13.  Given, 

14.  4. 

15.  12. 


LEAST  COMMON  MULTIPLE.— ART.  1O2. 


16.  Given. 

17.  36. 


18.  48. 

19.  90. 


20.  90. 

21.  240. 


22.  12600. 

23.  504. 


24.  1134, 

25.  144. 


REDUCTION  OF  FRACTIONS.—  ARTS.  120-4. 


1,  2. 

3.  i. 

4.  I- 

5.  -f. 
C.  i. 

7.  f. 

8.  $ 

9.  f 
10.  - 

11.  - 

12.  f. 

13.  « 

u.  - 


Given. 


i. 


16. 

18. 
21. 
22. 
23. 
24. 
25. 
26. 
27. 
28. 
29. 


4. 
5. 
2|. 

1. 

41|. 
30. 

28^. 
$28f. 


30. 

31,32.  Given. 

33.  -^. 

34. 

35. 

36. 

37. 

38. 

39. 

40. 

41. 

42. 

43. 


44. 

45, 46.  Given. 

47. 
48. 
49. 
50, 51~  Given 

52.  i-i 

53.  -4\. 

D  '1  •     ~90"  • 

55.  -f. 

56.  -Jy. 

57.  -jV- 

58.  T-fr. 


I36.J  ANSWERS. 

.REDUCTION  OF  FRACTIONS  CONTINUED. — A 


S5^V 


Axs. 


Ex. 


ANS. 


1-3.  Given. 

4.  -M-;  ffi; 

5.  -iVo- 


5          945    .   1440.  JULS_jQ..  JjlHi 
'•     T52  0>   2  52  0>  25  2  U>   25  2  0« 


7. 
s. 
9. 
10. 

nja_Z-fiJl_  •     1837  fi  .     1  0  5J 
•     1262  5  0   »      26250)      2626 
1  O          2  2.5.0  0     .      1  1  3  7  B  0  0   « 
J^'      625000   >        525UOIT- 

fern. 


13-15.  Given. 

IG.  if;  tt;  if. 
n.  if;  A;  ff. 
is.  fi;  ff  ;  M; 
19.  A;  W;  f&; 

'20.  ii;  f^;  ^;  if 

21.  -i4^-;  iVt;  T*T. 

22.     t2  0"  >     120  »    T2D~' 

23.  fn; 

24.  *H; 

25.  YW; 


ff. 


ADDITION   OF  FRACTIONS. 


ART. 
11-13.  Given. 

14.  2-fr. 

15.  1-H-. 


16. 

17. 
18. 
19. 


20.  2f||. 


21. 


23.    If. 


24. 
25. 
26. 

27. 


28. 
29. 
30. 


SUBTRACTION  OF  FRACTIONS. 


ART.  12§. 

11,13.  Given. 

14.  i. 

15.  i. 

16.  f 

17.  1. 


18. 
19. 
20. 
21. 
22. 
23. 


ART.  129. 

ART.  ISO. 

24,  25.  Given. 
26.  5i$. 

27.  H. 

31.  39|-. 
32.  1. 
33.  Hi. 

28.  I7£f. 

34.   2. 

29,  30.  Given. 

35.  0. 

MULTIPLICATION  OF  FRACTIONS.—  ARTS. 


11,12.  Given, 

13.  4, 

14.  10i. 

15.  6, 

16.  6. 

17.  18.  Given. 

19.  7*, 

20.  13f. 


2 1,22,  Given, 

23,  6, 

24,  18, 

25,  S8fr. 
28. 

27. 

28.  32^j. 


29,80,  Given,  40. 


81.  258. 
32,  889, 


J38, 84.  Giv^n,  43-45, 


85.  657, 
86, 


87, 88.  Given, 
39.  -A-. 


41. 
42, 


46,  +. 

47,  i>. 

48.  -jfr. 

49.  A- 
60.  Given, 


354 


ANSWERS.  [ARTS.  137-144 


EXAMPLES   FOR   PRACTICE. — ART.    13T. 


Ex.        ANS. 

Ex.         ANS. 

Ex.        AN.-. 

Ex.        ANS. 

1.  4cts. 

13.    112-£ctS. 

24.  fcf- 

35.  273  lets. 

2.  Gcwt. 

14.   235  p. 

25.  fcfr 

36.  61fM. 

3.  &9. 

15.   $16|. 

26.  $f£ 

37.  $4  If* 

4.  84-  bbls. 

16.  32  cts. 

27.  *H- 

38.  621-^cts. 

5.   10±c. 

17.  61i.  s. 

28.  1237|cts. 

39.  $8i£. 

6.  8|  a. 

18.  75  cts. 

29.  781-J-cts. 

40.   16|  s. 

7.  2|  s. 

19.   112icts. 

30.   115-1  cts. 

41.  391|cts. 

8.  5|  s. 

20.   2  16$  cts. 

31.  243^  cts. 

42.  652|  s. 

9.  $6}. 

21.  $56. 

32.  $3£. 

43.  &65i£. 

10.  $6f. 

22.   157icts. 

33.  $4££. 

44.  $138ff. 

11.  $12£. 

23.  $16*. 

34.  28i  s. 

45.  743f  m. 

12.   136  cts. 

4 


DIVISION  OF  FRACTIONS.— ART.  13§-143. 


11-1  3.  Given. 

22,  25.  Given. 

35.  f|. 

45.  323-^. 

14.  -ft. 

26.  22i. 

QA       Q    33 
OO4     *5"l;  36* 

46-49.  Given. 

15.  W. 

27.  -f. 

37.  7^V. 

50.  tt;  1-ft  ; 

16.  jj£ 

28.   IfB-. 

38.  6ff. 

lif  ;  2irh-; 

17.  iV*. 

29.  ||£. 

39.  3|f. 

li2?  5  i^f« 

18.  i. 

30,  31.  Given. 

40,41.  Given. 

52.  13^. 

19.  -&. 

32.  5|. 

42.  87i-. 

\  Q       1  J  B.  3. 

t)  0  .     1  "g  g  2  . 

20.  rh- 

33.   1-H-. 

43.   75i. 

54.  -f. 

21.    5  a  fi« 

34.  -ftV 

44.  212-f. 

55.  lif. 

EXAMPLES   FOR   PRACTICE. 

ART.  144. 

9.  5|1  Ibs. 

18.  $2f|f. 

26.  4ffH- 

1.  10  bu. 

10.  5-iB5  Ibs. 

19.  iiif  t. 

27.   l^Vi/W- 

2.  24  a. 

11.   10^  c. 

20.  87-flfr  s. 

28.  -sVV- 

3.   lli  Ibs. 

12.  St^bbls. 

21.  157-ftVb. 

29.  17|. 

4.  12  bu. 

•J  o       tf*O      B 

lo.    wt>  124. 

22.  9^. 

30.   1-ft. 

5.  4  gals. 

14.  7  cts. 

oq     a  A.  ft 

***«      475- 

31.  rh-- 

6.   14  A  yds. 

15     n  *4    g 

24.   185-Hf. 

QO        ,13  a 

o  ^  .    V?  o  5  * 

7«  5~H  yds. 

16.  ilt^r. 

25.   182iWr. 

<JQ       -iA*. 

oo  .     io7J» 

a,  10  ». 

17.  $6, 

ARTS.  162-1(35.] 

m 

DEDUCTION.— ARTS.  163-164. 


855 


IBBf'          ANS.                  ;Ex.            ANS.                   jEx.            AN». 

^1-6.  Given. 

39.  8i\r. 

73.  31|^C. 

7.  872  s. 

40.  5  m. 

74.   144  gals.  2  qts. 

8.  816  far. 

41.  4752000  in. 

75.  96  hhds.  17  gala. 

9.  JC4,  18s. 

42.  7650722  in. 

76.  7720  pts. 

10.  257s.  5d. 

43.  11  rn.  269-^  r. 

77.  40320  gi. 

11.  168000  far. 

44.  1585267200m. 

78.  17bbls.  13  gals. 

12.  81438  far. 

45.  180  qrs. 

79.  36  hhds.  6  gala. 

13.  £105,  4s.  8d. 

46.  636  na. 

80.  5428  qts. 

14.  £58,lls.7d.  If. 

47.  1620  na. 

81.  887  pts. 

15.  24462  far. 

48.  140  yds.  3  qrs. 

82.  17176  qts. 

16.  Given. 

49.  76  F.  e. 

83.  1681408  pts. 

17.  7200  grs. 

50.  260  E.  e.  2  qrs. 

84.  108  pks. 

18.  60144  grs. 

51.  480964  i  ft. 

85.  2675  bu. 

19.  2  Ibs.  loz.  12  p. 

52.  2472030  ft 

86.  1318140  sec. 

20.  4  oz.  9  p.  20  grs. 

53.  816000  a. 

87.  525960  min. 

21.  61bs.  loz.7p.2g. 

54.  94fV$V  sq-r- 

88.  31556928  sec. 

22.  Given. 

55.  466|  a. 

89.  11045160  min. 

23.  3650  Ibs. 

56.  437  a.  102  r. 

90.  1  57  h.  50m.  40  s. 

24.  171440  oz. 

57,  58.  Given. 

91.  850  w.  7  h.  36m, 

25.  23200  drs. 

59.  306  sq.  ft. 

92.  10305  mo.  3  w. 

26.  54  Ibs.  11  oz. 

60.  40  sq.  yds. 

5d.  16  h. 

27.  15cwt.2q.  151bs. 

61.  5  a.  2  r.  20  r. 

93.  7050  y.  8m.  3  w 

28.  6  Ibs.  12  oz. 

62.  24  a. 

5d. 

29.  3c.  2q.  4lb.  8oz. 

63.  129600  in. 

94.  34y.5m.lw.3d, 

30.  7  t  12  cwt.  3  q. 

64.  2557440  in. 

1  h.  46  m.  40  & 

10  Ibs. 

65.  562  T.  24  ft. 

95.  270000". 

31.  1  cwt.  4  Ibs.  6  oz. 

66.   129C.  56ft. 

96.  15300'. 

32.  Given. 

67.  8320  ft. 

97.  1296000". 

33.  44928  sc. 

69.  9288  in. 

98.  24°,  7',  40". 

34.  30  oz.  2  drs. 

70.  2160  ft 

99.  315s.  13°,  20'. 

35.  13  Ibs.  loz.  4d. 

71.  112  ft 

100.  231  s.  14°,  26' 

87.  356400  in. 

72.  3}  C. 

40". 

COMPOUND  NUMBERS  REDUCED  TO  FRACTION* 

ART.  165.       Y.  A  ?al.       13.  f£  m.         21.  iVir  hr. 

1,  2.   Given.     8.  i  hhd.       14.  fffr  1.        22.  y^nr  hr. 

3.  if-  s.             9.  -H  T.          15.  -H-  yd.       23.   ,4  Ib. 

4.  iibu.        10.  -iVir  cwt.   16-1  8.  Given.  24.  Wznr  T. 

5.  ii  pk.        11.  VW  qr.      19.  A  d.         25.  rfj  hbd 

0.  ftgal.       12.  ABU        20.  iWr*  d.  128.  -ft  gal 

556 


ANSWERS. 


.  166-171, 


FRACTIONAL  COMPOUND  NUMBERS?i§g 

REDUCED   TO  WHOLE   NUMBERS   OF   LOWER   DENOMINATION 


Ex.              ANS. 

Ex.              ANS. 

Ex.          ANS.                  ** 

ARTS.  166,  7. 

1.  Given. 
2.  12  s. 
3.  2s.  6d. 
4.  14s.  3}d. 
*.  5d,  14h.  24m. 

6.  9  h.  20  m. 
7.  1  m.  1  f.  24  r. 
8.  3  fur.  221  r. 
9.  Iq.lSlbs.  12oz. 
10.  8cwt.2q.7|lbs. 
11.  2pks.  fiiqi-. 

12.  2W-  qte. 
16.  T*&  d. 
17.  1-&  r. 
18.  tW  ft. 
19.  TWr  na. 
20.  Vi-lb. 

COMPOUND  ADDITION. 


ART.  168. 

9.  191b.llo.5p.23g. 

17.  92  bu.  3  p.  2  q. 

1,  2.  Given. 

10.  £70,  17s.  9d. 

18.  99m.  5  fur.  llr. 

3.  £19,  9s.  5d.  3  f. 

11.  15c.33lbs.9oz. 

19.  6hhds.  53g.3q. 

4.  £53,  5s.  5d. 

12.  1267  Ibs.  13oz. 

20.  8p.59g.2q.  Ipt 

5.  £58,  18a.  4d. 

13.  10T.1781.12oz. 

21.  109s.y.8f.  142  i. 

6.  451bs.4oz.2p.10g. 

14.  28yds.3qrs.  In. 

22.  3  la.  61  r.  48ft 

7.  3  Ibs.  7  oz.  12  p. 

15.  118yds.  3q.  2n. 

23.  99  cu.  ft.  227  in. 

8.  61  Ibs.  7  oz,  9  p. 

16.  65  bu.  1  pk. 

24.  73  C.  69ft  177  in. 

COMPOUND  SUBTRACTION. 

ART.  169. 

9.  6oz.  18  p.  2  grs. 

17.  8  yrs.  2m.  5  d 

1,  2.  Given. 

10.  13yds.  Iqr.  3na. 

16  h.  15  min. 

3.  £8,  7s.  lid.  2  far. 

11.  3yds.  2  qrs.  2  n. 

18.  Given. 

4.  £36,  3s.  7d.  2  far. 

12.  9  m.   18  r.    7  ft. 

6.  8T.  5c.2q.51bs. 

10  in. 

ART.  17O. 

6.  28  T.  17  cwt.  3qr. 

13.  54a.l49r.38s.f. 

19.  6  yrs.  4  m.  25  d. 

8  Ibs. 

14.  70  a.  Or.  33  r. 

20.  69  yrs.  1m.  2  Id. 

7.  9  gals.  1  qt.  3  gi. 

15.  128  ft.  1652  in. 

21.  3  yrs.  2  m.  23  d. 

8.  58hhds.  6g.  2q. 

16.  48C.  106ft.  58  in. 

22.  3  yrs.  7  m.  20  d. 

COMPOUND  MULTIPLICATION. 


ART.  171. 

1,  2.  Given. 

5.  £127,  12s.  6d. 
4.  £187,  14s. 

6.  £8,  9s.  3d.  3f. 

6.  £56,  6s.  3d. 

7.  £44,48. 


8.  7670  d.  2h.  4m. 

48  sec. 

9.  3lbs.  lo.  14p.  4g. 

10.  5  Ibs.  2oz.  8  p. 

11.  8T.  7  cwt.  91b. 

12.  5  T.  18  cwt.  2q. 

2  Ibs.  8  oz. 


13.  101  c.  I51b.7oa. 

14.  604gals.lq.2g 

15.  53m.  3  fur.  20  r. 

16.  212m.  6f.20r. 

17.  328  yds.  2  qrs. 

18.  96  a.  90  sq.  r. 

19.  603  sq,  yds. 


A.ETS.  173-192.] 


ANSWERS.  357 

COMPOUND   MULTIPLICATION   CONTINUED. 


r 


•"Ex.              ANS. 

Ex.            ANS. 

Ex.             ANS. 

?20.  18  C.  Ill  ft. 
21.  1512ft.  1064  in. 
22.  48«  23'  20". 
33.  1&4°  29'  52". 

24.  807  gals. 
25.  2452  gals.  2  qts. 
26.  £391,  14s.  5d. 
27.  3365  bu.  2  p.  4  q. 

28.  4937  yds.  2  qrs. 
29.  3012  T.  14  cwt. 
50lbs. 
30.  4687  bu.  2  pka. 

COMPOUND  DIVISION. 


ART.  173. 

8.  £6,  5s.  3d.  li  f. 

13.  §lb.  8foz. 

1-3.  Given. 

9.  £2,  2s.  6d.  2-^-f. 

16.  9yds.  2q.  l$n. 

4.  £2,9s.4d.2ff. 

10.  5oz.8p.  8g. 

17.  4m.  4  f.  17-J^-r 

5.  £5,  18s.  5d.  Iff. 

11.  lib.  3oz.  13  p. 

18.  19  bu.  2  qts. 

6.  £5,  7s.  Id.  3f  f. 

9f  grs. 

19.  2  a.  36f  f  r 

7,  £4,  15s.  4d. 

12.  10  Ibs.  llioz. 

ADDITION  OF  DECDIALS. 

ART.  187. 

7.  8.5284508. 

13.  330.967. 

1,  2.  Given. 

8.  19.57605. 

14.  10.709341. 

3.  320.67. 

9.  760.573. 

15.  2.0728. 

4.  2986.0501. 

10.  1310.9902. 

16.  0.408763. 

5.  81.271. 

11.  177.998. 

17.  0.607677. 

6.  111.9925. 

12.  33.4013. 

18.  0.7186423. 

SUBTRACTION  OF  DECIMALS. 

ART.  189. 

8.  10.69995. 

15.  55999.999001. 

1,  2.  Given. 

9.  0.23578. 

16.  0.675. 

3.  250.3905. 

10.  1.1011. 

17.  0.005994. 

4.  14.544. 

11.  1.400091. 

18.  0.3222. 

5.  13.25. 

12.  0.999999. 

19.  600.378. 

6.  144.96063. 

13.  130.8410699. 

20.  7855.999764. 

7.  0.875. 

14.  8897.319507. 

MULTIPLICATION  OF  DECIMALS. 

ARTS.  191,2. 

9.  0.50005.                18.  0.08568931. 

1.  231.41  yds. 

10.  50.1565195.          19.  0.00031275. 

2.  259.875  gals. 

11.  460.51.                  20.  0.0000022780402L 

3.  589.875  ft. 

12.  2650.1                   21.  0.0000025. 

4.  371.25  C. 

13.  5678.                     22.  0.00042. 

5.  519.675  r. 

14.  0.00187440781.    23.  0.001825. 

6.  474.6875.  m. 

15.  0.0024048072.      24.  0.00064125. 

7.  65365  Ibs. 

16.  0.000058175003.26.  0.000710U7S4. 

8.  44,8955  bbls. 

17.  0.0004000751. 

358 


ANSWERS. 


[Aps.  194-212. 

' 


DIVISION  OF  DECIMALS. 


Hx.                ANS. 

Ex.              AHS. 

Ex.            ANS. 

ARTS.  194,95. 

1.  6  coats. 
2.  9  loads. 
3.  12.3  days. 
4.  23.91  39+acres. 
5.  4.  5  rods. 
6.  3.15  barrels. 

7.  24.3936  days. 
8.  6.9  days. 
9.  15  boxes. 
10.  14.3. 
11,  12.  Given. 
13.  0.8. 
14.  0.001777+. 

15.  2.4. 
16.  10000. 
17.  500000& 
18.  17.6. 
19.  0.62. 
20.  31.7199+. 

REDUCTION  OF  DECIMALS. 


ART.  196. 

7.  0.8;  .8333+;  .1. 

11.  0.75m. 

1,  2.  Given. 

8.  0.16;  .4;  .04. 

12.  0.84375  Ibs. 

3.  f*. 

4        2. 

9.  0.625;  .4;  .05. 
10.  O.U25;  .0028+. 

ART.  201. 

*•      4' 

11.  0.025;  .003. 

1,  2.  Given. 

5.  f. 

12,  13.  Given. 

3.  7d.  2  far. 

6.  -?^-. 

4.  9s.  3d. 

7.  VV- 

ART.  20O. 

5.  3  qts.  &  .048  pts. 

8*.  |.° 

1,  2.  Given. 

6.  15hrs.  34.56  sec. 

v»      g  • 

Q     -A- 

3.  £.775. 

7.  3  qrs.  10  Ibs.  9oz. 

«'•    TT  • 

10.    fto  o  i)  0  &'• 

4.  £.625. 
5.  0.  75s. 

9.6  drs. 
8.  13cwt.  3qrs.  14lbs, 

ART.  19?. 

6.  0.625s. 

9.  3  pks.  &  .5248  pt 

1-3.  Given. 

7.  0.5  qts. 

10.  6  fur.  23  r.  3  yds 

4.  0.75;  .8. 

8.  0.  75  d. 

7.632  in. 

6.  0.15;  .28. 

9.  0.25  yds. 

11.  1R.  33+r. 

8.  0.375;  .2;  .6. 

10.  0.833+yds. 

12.  3qr.  &.  10096  na. 

ADDITION  OF  FEDERAL  MONEY. 
ART.  211. 


1.  Given. 

2.  $12.13. 
8.  $45.805. 


4.  $363.433. 

5.  $270.279. 

6.  $281.033. 


7.  $196.51. 

8.  $1022.529. 

9.  $76.121. 


10.  $216.728. 

11.  $317.207, 

12.  $10.545. 


SUBTRACTION  OF  FEDERAL  MONEY. 


ART.  212. 
1,  2.  Given. 
9.  $10.36. 


4.  $81.33. 

5.  $41.60. 

6.  $339.67. 


7.  $156.87. 

8.  $0.004-. 

9.  $0.174. 


10.  $54.422. 

11.  $100.088, 

12.  $900.066, 


.  215-23 


ANSWERS. 


'LICATION  OF  FEDERAL  MONEY. 


350 


n 


^£x.       ANS. 

Ex.         ANS. 

Ex.        ANS 

Ex.         ANS. 

^RT.  215. 
1-4.  Given. 

5.*jjfe 

6.  $i^P?5. 

7.  $4.6875. 
8.  $1.97625. 
9.  $4.1875. 

10.  $9.140625. 
11,  12.  Given. 
ART.  216. 
13.  $5.625. 
14.  $13.786875. 
15.  $77.46875. 
16.  $38.85425. 

17.  $57.09375- 
18.  $24.18. 
19.  $14.6875. 
20.  $18.375. 
21.  $142.50. 
22.  $13.005. 
23.  $127.50. 

24.  $1071.60. 
25.  $577.746. 
26.  $26.705. 
27.  $125.75088. 
28.  $36.2175. 
29.  $1071.00. 
30.  $8970.00. 

DIVISION  OF  FEDERAL  MONEY.— ART.  219. 


1-4.  Given. 
5.  21  Ibs. 
6.  $0.09375. 
7.  16qts. 
8.  25.5  Ibs. 
9.  24  w-m. 
10.  36  pen-ks. 

11.  25.142+q. 
12.  $3.50. 
13.  $1.8673+. 
14.  66  cords. 
15.  1.7894+b. 
16.  $0.07. 
17.  52  weeks. 

18.  $1.3698+. 
19.  $0.02. 
20.  $1.25. 
21.  465.55+b. 
22.  $10.2816. 
23.  $10.914. 
24.  $68.493+. 

ART.  220. 
1.  $17.770. 
2.  $12.95. 
3.  $21.485. 
4.  $123.07. 
5.  $1478.75. 
6.  $2305.625  n. 

PERCENTAGE.—  ART.  225. 

5-8.  Given. 

20.  $21. 

29.  $168. 

9.  $0.9021. 

21.  $1398. 

30.  $793.75. 

10.  $2.069075. 

22.  129.75  bx.  lost. 

31.  $43.375. 

11.  $0.96474. 

735.25  bx.  left. 

32.  $4  former. 

12.  $0.1809. 

23.  $63.333+. 

33.  $120. 

13.  $60.0451. 

24.  $106.8431. 

34.  0. 

14.  $300.0756. 

25.  $1.3332+. 

35.  $1720. 

15.  $450.168. 

26.  $468.75  lost. 

36.  $2152.50. 

16.  $13.952. 

$1031.  25  left. 

37.  $3100. 

17.  Given. 

27.  $1285.35. 

38.  $172.125. 

18.  $11.565. 

28.  $37.50; 

39.  $588.671875. 

19.  $58.8875. 

$951.5625. 

40.  1780  sheep. 

COMMISSION,  BROKERAGE,  AND  STOCKS. 


ART.  232. 

1.  Given. 
2.  $16.0068. 
3.  $21.51125. 
4.  $5.97  Agt.  ; 
$259.38  Ow. 
6.  $15.426+. 
£  $15.00, 

7.  $10.226+. 
8.  $70.993. 
9.  $54.37  com. 
$945.63  cot 
10.  $52.13  br. 
$10426  st, 
11.  $350. 
12,  $4£L575. 

13.  $521.93. 
14.  $29.27. 
15.  $406.437. 
16.  $3753.915. 
17.  $1759.308. 
18.  $477.65. 
19.  $7526. 
20,  $5000, 

21.  Given. 
22.  $527.50. 

23    $1275. 
24.  $3364. 
25.  $450. 
26.  $6750. 
27.  $598a 
28.  $450, 

300  ANSWERS.          [ARTS.  237-250. 

INTEREST.— ARTS.  237-241. 


Ex.              ANS. 

Ex.               Ana. 

Ex.          ANS. 

1-4.  Given. 

15.  $1080. 

27,  28.  Given. 

5.  $5;  $6;  $4;  $7. 

18.  $60  int.  ; 

29.  $65.166. 

6.  $2.118. 

$260  amt. 

30.  $7.4373* 

7.  $3.507. 

19.  $175  int.; 

31.  $9.20fcip 

8.  $3.465. 

$425  amt 

32.  $2.638. 

9.  $6.153. 

20.  $81.72  int.  ; 

33.  $7.526. 

11.  $9.817  int.; 

$422.22  amt 

34.  $29.043. 

$150.067  amt 

23.  $5.833. 

35.  $18.235. 

12.  $13.072  int.; 

24.  $5.629. 

36.  $206.58. 

$176.472  amt 

25.  $2.80  int.  ; 

37.  $4.125. 

13.  $24  int.  ; 

$62.80  amt 

38.  $2.916. 

$424  amt 

26.  $4.80  int.  ; 

39.  $5. 

14,  $535. 

$100.80  amt 

40.  $108.515. 

ART*.  244-246. 


1.  $.035. 

2.  $.04. 

3.  $.045;  $.05;  $.055. 

4.  $.07;  $.075;  $.09. 
6.  $.0015;  $.00366+; 

$.000666+;  $.002333+ 


7.  $.00166+;  $.00266+; 
$.00333+;  $.004;  $.0045; 
$.00466+. 


Given. 
9.  $.1583, 
10.  $.188, 


11-13.  Given. 

14.  $10.143. 

15.  Given, 


EXAMPLES   EOR   PRACTICE.  —  ARTS.    247-25O. 


1.  $1.81. 

2.  $5.021. 

3.  $1.642. 

4.  $0.916. 

5.  $13.904. 

6.  $242.346. 

7.  $391.613. 

8.  $42. 

9.  $224.193. 

10.  $675.863. 

11.  $898.88. 

12.  $1260.994. 


13.  | 

U08.616, 

14.  { 

12.815. 

15.  1 

(1022.25, 

16.  J 

11500. 

'7.  1 

13960.144. 

18.  $5125. 

19.  $1147.50. 

20.  $14.734. 

21.  $167.022. 

22.  $8635.505. 

23.  $16269.325, 

24.  $5.265. 

25.  $12.296, 

26.  $7.746. 

27.  $20.709, 

28.  $20.693. 

29.  $39.013, 

30.  $27.713, 

31.  $13.774. 

32.  $315.091. 

33.  $400.251. 

34.  $637.798, 


35.  $612.964. 

36.  $753.452. 

37.  $204.185. 

38.  $150.078. 

39.  $114.912. 

40.  $1382.33a 

41.  $4.00;  $4.5C 

42.  $0.36. 

43.  Given. 

44.  $366.66. 

45.  $426.4991 

46.  $780.07. 

47.  48.  Given. 

49.  £6,  12s.  4i 

50.  £10, 18s.  1  far. 

51.  £111,  13s.  4d, 

52.  £467,  9s.  3d 


ARTS.  253- 


ANSWEBS 


861 


PROBLEMS  IN  INTEREST.— ARTS.  253-255. 


1,  2.  Given. 
3.  6  per  cent. 


r  cent 

6.  7£  per  cent. 

7.  7i  per  cenc. 


fix. 


ANS. 


8.  5  per  cent. 

9.  9  per  cent. 

10.  5  per  cent 

11,  12.  Given. 

13.  $10000. 

14.  $11666.66f. 


ANS. 


15.  $17142.857^-. 

16,  17.  Given. 

18.  3£  years. 

19.  I6f  years. 

20.  14f  years. 

21.  10  years. 


COMPOf  TND  INTEREST.— ARTS.  257,  25§. 


1.  Given. 

2.  $91.866. 

3.  $348.207. 

4.  $335.024. 

5.  $1126.162. 

6.  $1351.791 


7.  $927.758 

8.  $2103.827. 

9.  $2123.198. 

10.  $4964.817. 

11.  $3195.818. 

12.  $26878.32. 


13.  Given. 

14.  $560.361. 

15.  $730.687  amt; 
$261.687  int. 

16.  $1524.468. 

17.  $4297.963 


DISCOUNT.— ARTS.  260-262. 


1,  2.  Given. 
3.  $443.925. 
4.  $153.508+. 
5.  $980.392+. 
6.  $18.293+. 
7.  $1674.4186+. 
8.  $1092.95+. 
9.  $28.4312+. 
10.  $27.8122+. 

11.  $4950.495+. 
12.  $1.698. 
13.  Given. 
14.  $5.979+. 
15.  $2.0625. 
16.  $8.75. 
17.  $1142.02. 
18.  $736.009+. 
19.  $41.9888. 

20.  $1276.173. 
21.  $4985. 
22.  $14985. 
23.  $1264.6173. 
24.  $15.1323+. 
25.  $17.6593+. 
26.  $59.5833+. 
27.  $69.231. 
28.  $457.944. 

INSURANCE.— ART.  265. 


1.  Given. 
2.  $6.5625. 
3.  $12.50. 
4.  $143.375. 
5.  $27.30. 
6.  $81.25. 

7.  $206.25. 
8.  $202.666. 
9.  $150. 
10.  $45.18. 
11.  $58.80. 
12.  $1950. 

13.  $573.75. 
14.  $2390. 
15.  $4205. 
16.  Given. 
17.  $16666.666. 
18.  $54545.455. 

20.  f  per  cent. 
21.  -J-  per  cent 
22.  7  per  cent. 
23.  Given. 
24.  $6793.478. 
25.  $11842.105 

PROFIT  AND  LOSS.— ARTS.  26T,  26§. 


2.  $6. 

3.  $5.79. 

4.  $43. 


5.  $56.25. 

6.  $250. 

9.  $26.1625. 


10.  $44.25. 

11.  $71.464. 

12.  $343.75. 


13.  $312.06. 

14.  $1163.75. 

15.  $29250* 


ANSWERS. 


PROFIT  ANI?   LOSS   CONTINUED. — A 


[ARTS.  269-294 


Ex.             ANS.              lEx.            ANS. 

Ex.              AN*.,1 

18.  50  per  ct. 

24.  4f  per  ct. 

30.  $20. 

19.  25  per  ct. 

25.  33i  per  ct. 

31.  $94.44$. 

20.  33i  per  ct. 

26.  16-4\  per  ct.  ; 

32.  $108.696. 

21.  33i  per  ct. 

$190. 

33.  $\2§M&&'.v 

22.  6f£.  per  ct. 

27.  23f£  perct.; 

34.  $373-^3-^-. 

23.  4-B-itf  perct. 

$2362. 

35.  $im.285. 

EXAMPLES   FOR   PRACTICE.  —  ART.  27O. 

1.  $1.98. 

8.  $.1395. 

14.  56-}-  cts. 

2.  16f  per  ct.  ; 

9.  30  cts.  per  gal. 

15.   12  cts.  per  yd 

40  cts.  gained. 

$37.80  gained. 

2  cts.  profit. 

3.  20  per  c.;  $3  g. 

10.  $1.062. 

16.  $1.008. 

4.  19^-  per  ct. 

11.  $1383.75  lost. 

17.  45  cts. 

$6  gained. 

$9.3275  bbl. 

18.   14f  per  ct. 

5.  $2.30. 

12.   100  per  ct. 

19.  21-HB-  perct. 

6.  $4.79i. 

$37.50  gained. 

20.  $.9375  per  a. 

7.  $5.60. 

13.  $450. 

$3125  lost. 

DUTIES.— ARTS.  273-274. 


1,  2.  Given. 

3.  $87.75. 

4.  $202.50. 

5.  $941.60. 

6.  $8640. 


7.  Given. 

8.  $75. 

9.  $71.224. 

10.  $131.6525. 

11.  $93. 


12.  $69.0375. 

13.  $186.4625. 

14.  $850. 

15.  $1000. 

16.  $1695. 


17.  $1504.80 

18.  $2898. 

19.  $3592.75. 

20.  $4500.375. 

21.  $2819.126 


ASSESSMENT  OF  TAXES.— ARTS.  27§-279. 


1,  2.  Given. 
3.  $12.25. 
4.  5  cts.  on  $1. 
$57.50  man's  t. 
5.  $76.50. 
6.  2  cts.  on  $1. 
$102.40.  C's  t. 

7.  $71.20D'st. 
8.  $309  G's  t. 
9,  10.  Given. 
11.  $30.22  B'st. 
12.  $117.51  C'st. 
13.  $159.47  D's  t. 
14.  $285.22  E's  t. 

15.  $510.50  F's  t. 
16.  $405.25  GW. 
17.  $307.70  H's  t. 
18.  $661  J's  t. 
19.  $300.51  K's  t. 
20.  $90.75  L'st. 
21.  $612.25  M'st 

PROPERTIES  OF  NUMBERS.— ARTS.  2 §6-2 94. 


1.  Given. 
2.  27  cows. 
3.  185  acres. 
5.  $36. 
6.  387  sheep. 

7.  Given. 
8.  24  years. 
9.  1050  fern. 
11.  lOyrs.; 
15  yrs. 

12.  $36  tea; 
$27  molas. 
14.  10  yrs. 
15.  8  rods. 
17.  $825. 

18.  1350  ap. 
20.  12  sailors, 
21.  8  flocks. 
23.  7  years. 
24.  12  mark 

ARTS.  298-303. 


ANSWERS. 


363 


AN. 


ALYSIS.—  ART.  298-303. 


Ex, 

ANS. 

Ex. 

ANS. 

Ex.        Ans. 

Ex.        ANS. 

1. 

.Given. 

18. 

306  m. 

37;  $85.50. 

57..69A-t- 

V2./ 

"8565.41." 

80*  m. 

38.  $75.74+. 

58.  7-^r  cts. 

$22.20. 

21. 

141  ds. 

39.  $1.00. 

59.  125.8  bu. 

86  ao 

22. 

57f  ds. 

41.  $31.50. 

60.  50  cts. 

23. 

54  ds. 

42.  $2.85. 

63.  $420. 

6. 

$204. 

24. 

32!  mo. 

44.  £1. 

64.  $90. 

7. 

$30. 

25. 

288  ds. 

45.  $53333!. 

65.  $120. 

8. 

$166.40. 

26. 

70  cts. 

46.  $93i. 

66.  $95. 

9. 

$47.50. 

27. 

$4.20. 

47.  $23.125. 

67.  £l83f. 

10. 

$97.50. 

28. 

36  cts. 

48.  60  ds. 

68.  £472i. 

11. 

$4.50. 

29. 

$5.22. 

49.  100  ds. 

69.  £11250. 

12. 

$12.12. 

31. 

$0.96?. 

50.  50  days. 

70.  $252.35. 

13. 

$4.161. 

32. 

47is. 

52.  64  bu. 

71.  $30. 

14. 

$3.50. 

33. 

$25.60. 

53.  2  cords. 

72.  $250. 

15. 

$2.05. 

34. 

$3. 

54.  270  pair. 

73.  $240. 

16. 

108!  m. 

35. 

20  cts. 

55.  199f  Ibs. 

74.  £337i. 

17. 

112njfbu. 

36. 

$18.04. 

56.  98-i53lb3. 

75.  £1275. 

77. 

$133.33!,  A. 

800  b.,  B.  83.  661  cents. 

$166.661,  B. 

1000  b.,  C.         $200,  1st. 

78. 

107!  bu.,  A. 

533ib.,  D.         $266.661,  2d. 

85$  bu.,  B. 

81.  $315,  A.             $333.33!,  3d. 

57!  bu.,  C. 

$525,  B.      84.  80  cents. 

79. 

$600,  A. 

$420,  C.      85.  $64.1379tr£&,  A. 

$375,  B. 

82.  $1250,  X.            $105.1  m-foVm  B. 

$525,  C. 

$1750,  Y.           $147.7488iHf»C. 

80. 

6661bbls.A. 

$2000,  Z. 

86. 

10  cts. 

Q1         1  K:    S3  1)  B 
«/  1  .      J.  O  2  0  4  S  if  • 

102.  20  men. 

112.  60. 

$500,  B. 

1 

&644.15,M. 

103.  7!days. 

113.  24  ft. 

87. 

$.042+. 

95. 

6  shil. 

104.  2i  mo. 

114.  $136. 

88. 

$.ll!|-. 

96. 

4  cts. 

105.  720  m. 

115.  $14400, 

89.. 

100  b.,  A 

97. 

$22.50. 

106.  224  bu. 

116.  72  yrs. 

661b.,B 

40!icts 

108.  240  s. 

117.  48. 

33  £  b.,C 

98. 

65ff  cts. 

109.  $288. 

118.  72  sch. 

BO. 

10  per  ct 

99 

9  ?gf  cts. 

110.  1440  m 

119.  $15600. 

$1500,  A 

100.  17-^0. 

111.  144. 

120.  60  tree*. 

364 


ANSWERS.          [ARTS.  3 277-362, 


SIMPLE  PROPORTION.— ARTS.  3 


Ex.        ANS. 

Ex.        ANS. 

Ex.       ANS. 

Ex.             •AW>JV 

1-3.  Givep. 

15.  $6.131. 

26.  18f  bbls. 

37.  80  cents?* 

4.  $100.    * 

16.  Given. 

27.  $60. 

38.  65£  we 

5.  $75. 

17.  $784. 

28.  75  feet. 

39.  £56,  13s.  4d. 

6.  $30.75. 

18.  $216. 

29.  100  days. 

40.  £186,  2s.  4i<J 

7.  1140  bu. 

19.  $515. 

30.  105  a. 

41.  3s.  9fod. 

8.  240  miles. 

20.  £22,  10s. 

31.  $595. 

42.  £585,  Is.  4^ 

9,  10.  Given. 

21.  1440m. 

32.  130fcwt. 

43.  £3,  12s.  6d. 

11.  12  days. 

22.  £2,  5s. 

33.  133f  sp. 

44.  £41,  12s.  6d 

12.  $59.50. 

23.  $1.70+. 

34.  $1500. 

45.  3|  hours. 

13.  13$  mo. 

24.  $3.75. 

35.  $25U. 

46.  5  min. 

14.  $7. 

25.  $5000. 

36.  362  days. 

47.  12  hours. 

COMPOUND  PROPORTION.— ART.  331. 


1-4.  Given. 

5.  96  men. 

6.  10  men. 


7.  6  days. 

8.  74-    days. 

9.  170f  bu. 


10.  80  days. 

11.  6  men. 

12.  9  months. 


13.  $18. 

14.  £384. 

15.  90  day* 


DUODECIMALS.— ART.  336. 


1.  Given. 

2.  46ft.  10  in.  6". 

3.  13  ft.  7  in.  2". 

4.  82  ft.  9  in.  4". 

5.  210ft.  4  in.  6". 

6.  1364ft.  3  in. 

7.  149  ft.  5  in.  6" 


8.  137ft.  2  in.  8" 

9.  35  ft.  6  in.  8".  6'", 

10.  38ft.  2  in.  4". 

11.  82ft.  5in.8".4'". 

12.  86  ft. 

13.  210ft.  4  in.  6". 

14.  2200  ft. 


15.  6375  ft. 

16.  472ft.  6  in. 

17.  484ft.  lin.  9".  4"' 

18.  8100  bricks. 

19.  $22.50. 

20.  $3.555£. 


SQUARE  ROOT.— ARTS.  351-359. 


1,  2.  Given. 

10.  111. 

3.  25. 

11.  232. 

4.  30. 

12.  729. 

5.  35. 

14.  1.4. 

6.  42. 

15.  5.4. 

7.  54. 

16.  15.3. 

8.  69. 

17.  .35. 

9.  93. 

18.  .881. 

19.  1.4142+. 
20.  4.123+. 
21.  13.228+. 
22.  342. 
23.  3212. 
24.  f  . 
25.  if  =£. 
26.  |-=2i. 

27.  -V-=7i. 
29.  10yds. 
30.  50  miles. 
31.  200  miles 
32.  60  feet. 
33.  24.97+ft. 
34.  42.426+r 
35.  56.568+r> 

CUBE  ROOT.— ART.  362. 


1,  2.  Given. 

3.  12. 

4.  24. 


5.  72  in. 

6.  83  ft. 

7.  125ft. 


8.  1.25-f. 

9.  1331. 
10.  2.3. 


11    4.5. 

12.  *. 

13.  *. 


ARTS,  367-406.  j          ANS  $i;VsV   ^  <   -    \K  V    365 
EQUATION  OF  PAYMENTS.—  ART?  36%^*  *l  *} 

3.  6  ra                •>.  4  months.      7.  1£  yrs.     *> 

69.«6^  mo. 

''•      q^Effifc  6.  6  months,      8.  3  months. 

VI  * 

rf^i  yrs. 

^KJBLTNERSHJP.^ARTS.  369,  i 

F&. 

I.  Given. 

L  $58&4te-izIaj-,    A.          $3333.33i,B's. 

2.  snra'ovAV 

$112>.8W-M-,  B.          $4000,  C's. 

$1800,  B's. 

$1670.175^,  C.     8.  $100,  A's. 

$2000,  C's. 

$2210.526^,  D.          $120,  B's. 

8.  $120,  A's.   , 

5.  $300,  A's.                      $120,  C's. 

$160,  B's. 

$400,  B's.                 9.  $30  apiece. 

$200,  C's. 

$600,  C's.                10.  $40.019^,  A. 

$700,  D's.                      $ 

oo  nhh  i  o  T     "D 
00,4  1  i~Jo5",  Of 

6.  $2666.661,  A's.            $ 

1  1  *7  '7AQ_Z_3_  O 
1  1  I  .  i  UOj^  {)  9  ,w. 

EXCHANGE  OF  CURRENCIES.—  ARTS.  3T5-3T9. 

1,  2.  Given.      8.  £300.               14.  £240. 

20.  $640.625. 

3.  $484.          10.  $6391.855.       15.  £250. 

21.  $790.93$. 

4.  $1334.63.    11.  Given.               17.  $162.50. 

22.  $1147.73f 

5.  $2179.815.  12.  £113,  8s.          18.  $244. 

23.  $2436.428. 

7  £100,  10s.  |13.  £l86,3s.7.2d.  19.  $252.25. 

24.  $4003. 

MENSURATION.—  ARTS,  3§O-4O6. 

1.  Given.             '18.  Given. 

36.  152  sq.  ft. 

2.  640  a.             U9.  11309.76  sq.  r. 

37.  Given. 

3.  26  a.  65  r.       20.  2037.18496  yds. 

38.  21205.8  cu.  ft. 

4.  50  rods. 

21.  8  r. 

40.  30000  sq.  yds. 

5.  80  rods. 

22.  16  ft. 

42.  123J  cu.  ft. 

6.  Given. 

23.  Given. 

44.  84 

so.  yds. 

7.  4  a.  75  r. 

24.  24  feet. 

45.  Given. 

8.  Given.  * 

25.  160  rods  long; 

46.  45945.75+  cu.  ft. 

9.  558  rods. 

80  rods  wide. 

48.  4084.067  sq.  ft. 

10.  Given. 

26.  12. 

49.  Given. 

11.  49  14  sq.ft. 

27.  84. 

50.  201061760  sq.  m. 

12.  Given. 

28.    £. 

51.  904.77792  cu.  in. 

13.  62.35-|-yds. 

30.  5425  cu.  ft. 

52.  26808234666Gf  in. 

14.  Given. 

31.  27  feet. 

53.  Given. 

15.  314.159  rods. 

32,  33.  Given. 

54.  6  in. 

16.  Given. 

34.  9200  cu«  ft. 

56.  12.599-fft* 

17,  200  yds. 

«5,  Given. 

58.  244,346+  gal* 

MISCELLANEOUS  EXAM 


Ex.              AKS. 

Ex.^WANs.  M 

Ex.              AMS. 

I.   $1250.  m#  c-f? 

2.  $90.     * 

49.  7812.5  IDS.* 
50.  5082.      /^ 

95.    |MBl 

3.  $1  29.374.  /L^ 

51.  6776.    ///y 

96.  7  Ibs.  l-SrHfcz 

4.  $H069.     /7«7 

52.  $322.50.       , 

97.  $13440. 

5.  $600.031}. 

53.  $124.  $U  ** 

98.  58  sheep. 

6.  $3332.531}. 

7.   1163  bbl=>- 
9.  500  saddles. 

54.  $80  6t>\\  / 
55.  $256.     f*  W 
56.  $436.50. 

99.  72^days. 
100.  313ipJaysT 
101.  6270  days. 

10.  50  horses. 

57.  $158. 

102.  12  days. 

12.  $142.45. 

58.  £212. 

104.  420  bu. 

13.  $211.05. 

59.  £193,  10a» 

105.  120  days. 

14.  $20.344. 

60.  £114. 

106.  $7266.66f. 

15.  $18.659. 

61.  £219. 

107.  $12170.31f. 

16.  $1023.667. 

62.  £61,  5s. 

108.  $15280.83$. 

17.  $4662.031. 

64.  £155. 

109.  £645,  9s.  44d. 

18.  $10446.33i. 

65.  $73. 

110.  36  rods. 

19.  $1318.84^-. 

66.  $1250. 

111.  160  rods. 

20.  $1531.396JL-2' 

67.  300  tons. 

112.  80  rods. 

22.  $19.294. 

68.  £11,  5s. 

113.  600  sq.  yds. 

24.  $6090. 

69.  $26.16. 

114.  2400  sq.  rods. 

25.  $106.25. 

70.  $65f. 

115.  63. 

26.  $406.625. 

71.  $506.  13i- 

116.  264. 

27.  $13.885. 

72.  £1,  10s.  8}d. 

117.  ^-. 

28.  $396.625. 

73.  lib.  loz.  22grs. 

118.  ff. 

29.  $579.072. 

74.  1  oz.  18}  pwts. 

119.  144rk.;48fii« 

30.  $846. 

75.  11s.  4d. 

120.  18  inches. 

31.  $987.75, 

76.  £9,  12s. 

121.  U  yds. 

Q  O         tft»  A    £*  r\  1 

O^y.     tJpv/.UJg"« 

77.  $14.615. 

122.  12  yds. 

34.  $0.094. 

78.  $17145. 

123.  60  yds. 

35.  20  per  ct.;  $9  12. 

79.  £4633^. 

124.  24  yds. 

36.  50  per  ct. 

80.  110  yds. 

125.  $72. 

37.  44^  per  ct. 

81.  1980  gals. 

126.  $300  A's  sh. 

38.  $3478.667. 

82.  $68.649. 

$420  B's  sh. 

39.  $5557.68. 

83.  78  Ibs.  2  oz. 

$800  C  put  in 

40.  $8724.375. 

85.  m  shil. 

127.  $720  A's  sh. 

41.  3904  ;  4654. 

86.  £4,  8s.  8d.     • 

$  1200  B's  sh. 

42.  2759  ;  2884. 

87.  331  cts. 

$1680  C's.  sh, 

43.  428.                      88.  220^  Ibs. 

128.  1  hr.  5-ft-  mia 

44.  50.                        |89.  169-yV  a. 

1129.  1200  sheep. 

91.   1290.48  bu.           130>  isfrft.    * 

48.  100.                    ;93.  105ft.                  <132.  34hrs.;323m 

fHOS 


EECOMMEN 

PRACTICAL  ARITHMETIC. 


j^of  Teachers,  Superintendents,  Trustees  and  School 
L  is  respectfully  incited  to  the  following  Recommendation* 
cockers  ard.  School  Committee  of  New  Haven,  of  Thorn- 


PRACTICAL    ARITHMETIC. 

From  A.  D.  Stanley,  A.  M.,  Professor  of  Mathematics  in  Yale  College 
From  such  an  examination  as  I  have  been  ablo  to  make  of  Thomson* 
'•  Practical  Arithmetic,"  I  cannot  doubt  that  it  will  hold  a  high  rank  as  an  el- 
ementary work  in  our  Academies  and  Schools.  It  will  commend  itself  to 
teachers  for  the  clearness  and  precision  with  which  its  rules  and  principles  are 
stated,  for  the  number  and  variety  of  examples  it  furnishes  as  exercises  for  ths 
i>upil,  and  especially  for  the  care  which  the  author  has  taken  to  present  ap- 
propriate suggestions  and  observations  wherever  they  are  needed,  to  clear  up 
any  difficulties  that  are  likely  to  embarrass  the  learner.  In  recommending  the 
work  a;  a  class-book  for  pupils,  it  is  not  unimportant  to  state,  that  the  author 
has  himself  had  much  experience  in  the  business  of  instruction,  and  has  thus 
had  occasion  to  know  where  there  was  room  for  improvement  rh  the  elemen- 
tary treatises  in  common  use.  Without  such  erperionce,  no  one  c;m  be  quali- 
fied to  prepare  a  class-book  for  schools.  A.  D.  STANLEY 
Vale  College,  Dec.  4,  1846. 

We  cordially  concur  in  the  views  expressed  by  Prof.  Stanley,  respecting 
Thomson's  Practical  Arithmetic. 

AZAR[AH  ELDRIDGE,  A.  M.,  Tutor  In  Nat.  Philosophy. 
JOSEPH  EMEUSON,  A.  M.,  Tutor  in  Mathematics. 
SAMUEL  BRACE.  A.  M.,  Tutor  in  Greek. 
JAMES  HADLEY,  JR.  A.  M.,  Tutor  in  Latin. 
EDWARD  C.  HER  RICK,  A.  M.,  Librarian. 
HAWLEY  OLMSTEAL),   A.   M.,   Principal   of   Hopkln*' 
Grammar  School.  [for  Boy« 

A.  N.  SKINNER,  A.  M.,  Princ.  of  Select  Classical  School 

From  Stiles  French.  A.  M.,  Teacher  of  Mathematics. 

I  have  examined  Mr.  Thomson's  new  Practical  Arithmetic,  with  careful  «»< 
tention,  and  have  decided  to  adopt  it  for  my  classes  of  beginners. 

To  the  teachers  of  our  common  schools,  this  Arithmetic  may  be  particularly 
lecommended,  as  HI  all  respects  convenient  and  exctllent  for  their  use. 

New  Haven,  Dec.  5,  1845.  STILES  FRENCH, 

From  the  examination  which  I  have  been  able  to  make  of  the  Practie.il 
Arithmetic,  by  J.  B.  Thomson,  A.  M.,  I  coincide  fully  in  the  recommendatioa 
ofstby  Mr.  French,  to  whim  the  department  of  mathematical  instruction  in 
our  irstitate  is  more  immediately  intrusted. 

WM.  II.  RUSSELL,  Principal  of  the  Collegiate  an# 
Commercial  Institute,  New  Haven. 

Yf9  fully  concur  in  the  above  recommendations. 

AMOS  SMITH,  Principal  of  S«lect  School  fbr  Boj*. 
B.  W  COLT, 


The  publishers  have  the  satisfaction  of  announcing  that  the  ~Boardvf  Schte* 
Vititora  have  unanimously  adopted  Thomson's  "  Practrcal-Atithmeti'cV  fol 
the  use  of  the  Public  Schools  in  the  city  of  New  Havenj\| 

"  At  a  meeting  of  the  Board  of  School  Visitors  fi^^he  First  School  Society 

SC"certified  by     '  ALFJRED  

H.  G.  LEWIS,  Secretary.  ^j|f' 7^  'V 

From  the  Hon.  Judge  Blackman,  A  M.,  Chairman  of  the  Boa&toASchool  Vis 
itors  of  the  City  of  New  Haven.        4B 

James  B.  Thomson,  Esq.,— Dear  Sir,— I  have  examined  w"iWsf5me  atten- 
tion your  "  Practical  Arithmetic,"  and  consider  it  decidedly  the  best  work  for 
inculcating  and  illustrating  the  principles  and  practice  of  Arithmetic,  which  I 
have  ever  seen.  Your  illustrations,  in  the  form  of  problems  to  be  solved,  are 
drawn,  in  a  great  measure,  from  the  familiar  scenes  of  early  life  ;  and  while 
the  young  learner  is  interested  in  the  solution  of  problems  which  he  feels  are 
practicable,  he  is  encouraged  to  persevere  in  a  study  which  would  otherwise 
be  dull  and  forbidding,  and  is  thus  imperceptibly  led  to  acquire  and  understand 
the  rules  of  arithmetic,  which  he  now  knows  to  be  true. 

I  am  glad  you  have  removed  "  the  ancient  landmarks"  of  common  school 
"  ciphering,"  and  thus  permitted  a  child  to  understand  what  he  reac's  ;  instead 
of  torturing  his  mind  with  a  jargon  of  words  which  he  cannot  understand,  and 
requiring  him  to  work  by  a  rule  which  he  cannot  explain. 

I  need  hardly  say,  that  the  inductive  method  which  you  have  adopted,  ii 
decidedly  the  most  philosophical  and  intelligible  mode  of  acquiring  a  knowledge 
of  arithmetic ;  and  as  such  I  shall  cheerfully  recommend  your  work  for  gen 
eral  use  in  the  schools  of  this  city. 

I  ought  not  to  overlook  the  copious  references  by  which  your  rules  are  ex 
plained,  anffthe  mind  of  the  student  assisted  in  his  labors  ;  nor  the  skill  with 
which  the  publishers  have  executed  their  part  of  the  work. 

I  am,  dear  Sir,  very  respectfully  yours,  ALFRED  BLACKMAN. 

Nov.  29th,  1845. 

From  the  Principals  of  the  Publit  Schools  in  the  City  of  New  Haven. 

New  Haven,  Nov.  28th,  1845. 

I  have  given  Thomson's  "  Practical  Arithmetic"  as  careful  a  perusal  as  my 
time  would  permit.  I  think  it  a  work  of  very  great  merit.  The  plan  of  it, 
which  has  been  ably  carried  out,  appears  to  me,  to  be  natural  and  philosophi- 
cal. The  definitions  and  rules  are  exceedingly  clear,  and  will  be  easily  under- 
stood by  those  for  whose  instruction  they  are  designed.  The  notes  and  obser- 
vations,  which  frequently  occur,  are  admirably  condensed,  and  afford  much 
valuable  aid  and  information.  The  examples  for  both  mental  and  slate  exer 
cises,  are  appropriate  and  abundant,  and  while  the  former  are  sufficiently 
simple  to  make  the  principles  clear  to  the  tyro's  mind,  the  latter  will  secure 
sufficient  practice  with  the  pencil,  to  fix  them  there.  I  notice  in  almost  every 
new  rule,  suggestion  and  illustration,  that  the  pupil  is  pointed,  by  the  means  of 
numbers  in  brackets,  to  principles  he  has  already  studied  ;  this  is  an  excellent 
plan ;  it  will  be  found  highly  useful  to  him,  and  very  convenient  to  his  in- 
structor. I  will  not  attempt  to  make  allusion  to  all  the  peculiarities  and  ex- 
cellencies of  the  work ;  suffice  it  to  say,  that  I  consider  it  the  best  of  all  the 
excellent  works  of  a  similar  kind  with  which  I  am  acquainted.  1  shall,  with- 
out delay,  request  the  sanction  of  the  Board  of  Visitors,  for  its  adoption  in  the 
school  under  my  care.  J.  E.  LOVELL,  Principal  of  the 

—  Lancasterian  School. 

We  fully  concur  in  Mr.  Lovell's  views  respecting  Thomson's  "  Practica1 
Arithmetic,"  and  are  gratified  to  know  that  the  Board  of  Visitors  have  adopt 
ed  it  for  the  Public  Schools  of  this  city. 

PRELATE  DEMICK,  Principal  of  Whiting  st.  School. 
WM.  H.  WAY,  Principal  of  Wooster  st.  School. 

Ftom  R«#.  J.  JBrcwert  A.  M.,  Prln.  of  Elm  st.  Female  Seminary,  New  Haven 
Owur  Sixr-After  Mai  of  a  number  of  different  work*  which  ha\%  boea 


brought  to  my  notice,  >  I  have  concluded  to  adopt  your  Practical  Arithmetic  hi 
my  Seminary.  SBesiael  other  and  higher  merits  which  those  more  exclusively 
devoted  to.  mathematical  pursuits  will  be  ready  to  point  out,  the  following  are 
excel  1<  ev&p-  experienced  teacher  will  be  able  to  appreciate. 

1.  I\  umbering  bf-ffie'  articles,  by  which  one  may  readily  refer  to  any  pro- 
\\o\i-  step.  '         •  «. 

2.  Invariab'lyigiyin!:  the  important  definitions  and  general  rules  in  Italics. 
Sr^wwing-jhto  smaller  type,  in  the  form  of  Notes  and  Observations,  the 

^literature"  Qijhe  subject,  and  useful  hints  for  teachers  and  advanced  pupils 
Neyy  Haven,  Dec.  5,  1845.  JOSIAH  BREWER. 


,  A.  M.,  Principal  of  the  Young  Ladies'  Institute, 

New  Haven,  Ct. 

>  Mr.  Thomson  —  Sir,  —  In  teaching  Arithmetic,  I  have  been  exceedingly  em- 
barrassed in  deciding  upon  a  text  book  for  my  pupils,  but  am  now  happy  to  find 
inks  difficulty  removed.  I  can  confidently  recommend  your  Practical  Arith- 
metic, as  combining  excellencies  to  be  found  in  no  other  elementary  work  on 
this  subject.  In  the  lucid  and  natural  arrangement,  the  analysis  of  principles, 
and  the  full  explanation  of  each  step  as  you  proceed,  it  exhibits  many  traces 
of  the  skill  which  appears  in  the  other  parts  of  your  Mathematical  Series  al 
ready  published.  Yours,  truly,  VVM.  WHITTLESEY 

Woolsey  Hall,  New  Haven,  Nov.  26,  1845. 

From  E.  L.  Hart,  A.  M.,  Principal  of  English  and  Classical  School  for  Boys. 

Messrs.  Durrie  &  Peck,  —  I  have  carefully  examined  Thomson's  Practical 
Arithmetic,  and  fully  believe  that  it  is  superior  to  any  other  Arithmetic  now 
before  th«  public.  1  like  it  for  its  excellent  arrangement  —  for  its  very  clear  il- 
lustration and  exposition  of  principles  —  for  its  accuracy  in  tables  of  weights 
and  measures,  some  of  which  are  incorrect  in  all  other  Arithmetics  with  which 
I  am  acquainted—  and  for  its  eminently  practical,  business-like  character.  I 
shall  introduce  it  into  my  school  as  soon  as  it  is  practicable.  Yours,  &.c. 

New  Haven,  Nov.  27,  1845.  EDWARD  L.  HART. 

From  J.  D.  Farren,  Esq.,  Principal  of  Select  School  for  Boys. 

Mr.  J.  B.  Thomson  —  Dear  Sir,—  I  have  examined  your  Arithmetic,  and  must 
gay  I  am  very  highly  pleased  with  jt.  Its  merits  will,  at  once,  present  them- 
selves to  the  mind  of  every  one  \*ho  will  examine  it.  The  thorough,  syste- 
matic course  pursued,  is  a  grand  one.  I  have  introduced  it  into  my  school, 
which  is  more  in  its  favor  than  anything  I  can  say. 

I  would  say  to  those  teachers  who  prefer  to  have  their  pupils  work  by  the 
light  of  the  sun  rather  than  that  of  the  moon,  use  Thomson's  Arithmetic. 

JOSEPH  D.   FARREN. 

New  Haven,  Nov.  28,  1845. 

From  5.  A.  Thomas,  Esq.,  Principal  of  New  Haven  Practical  School  for  Boys. 

NEW  HAVKN,  Dec.  1,  1845. 

Mr.  J.  B.  Thomson  —  Sir,  —  From  the  examination  which  I  have  been  able  to 
give  your  "Practical  Arithmetic,"  I  think  it  a  valuable  addition  to  that  class 
of  School  Books.  It  contains  many  useful  improvements,  both  in  arrangement 
and  in  matter.  The  arrangement  of  subjects  is  decidedly  the  best  I  have  seen 
—  the  examples  are  judiciously  selected  and  arranged,  and  the  explanations  and 
rules  clear  and  concise.  I  think  the  work  well  calculated  to  lead  the  pupil  to 
EU  easy  and  rapid  acquisition  of  the  science  of  Arithmetic. 

Yours,  truly,  S.  A.  THOMAS. 

From  J.  H.  Rogers,  Esq.,  Principal  of  Fair  Haven  Family  Boarding  School. 

After  a  thorough  examination  of  Thomson's  Arithmetic,  I  believe  it  to  be 
superior  to  any  other  extant.  It  is  sufficient  to  say  that  I  have  adopted  it  in 
my  school.  The  mathematical  labors  of  Prof.  Thomson  evince  the  erudition 
of  a  ripe  scholar,  united  with  the  skill  of  a  practical  teacher.  I  have  tested 
the  value  of  his  Algebra  and  Geometry,  in  my  school,  with  great  satisfaction, 
and  have  no  doubt  his  Arithmetic  will  fully  sustain  his  high  reputation  as  an 
author.  J.  H,  ROGERS. 

Fair  Raven,  Nov.  28,  1845. 


Tr«m  Wm,  B.  Oretne,  A.  B.,  Principal  of  Millbrd 

klt-woRD,  Nov.  29,  1,845. 

J.  B.  Thomson,  ,  Esq.  —  Dear  Sir,  —  I  have  examined  your  Practical  Arithme 
Me,  and  am  pleased  to  observe  the  clearness  and  preci-sio'rrwUla  vvl>"w.h  the  sub 
ject  is  presented  —  the  same  that  have  so  highly  clwracteTr/ert^yimr  Algebra 
and  Geometry,,  and  so  happily.  adapted  them  to  the  capacities  of  the  young.— 
.Such  a  work  has  long  been  needed  in  our  schools  and  acadeimtis.  It  meuts  my 
views  so  well,  that  I  have  introduced  it  into  my  schooldMfl 

Yours,  truly,  cr^vEENfot 

From  /.  O.  Hobbs,  A.  M.,  Principal  of  Washington  InstLtale,  ,J$£iy  York  City. 

Gentlemen,  —  1  have  carefully  examined  Mr.  'I'homsoa^j^HEicnl^Arithiiie- 
tic,"  and  do  most  heartily  add  my  testimonial  to  those  alreHdy  gtven  in  its  fa- 
vor. It  is  indeed  a  work  of  very  great  merit,  comprising  many  excellenciesTo, 
a  small  compass.  Its  value  as  a  practical  school-book  will  be  more  apparenS 
on  a  second  and  thorough  examination.  While  as  an  elementary  work  it  de- 
serves the  place  in  our  best  schools  that  is  occupied  by  the  best,  /  know  of  no 
other  so  wel!  adapted  to  general  use.  ISAAC  G.  HOBBS. 

New  York,  Aug.  1,  Iri4t5. 

From  the  Teachers  of  the  Normal  School  connected  with  the  Public  School*  of 

the  City  of  JVe/o  York. 

Thomson's  "Practical  Arithmetic"  is  an  exceedingly  well  arranged  bonk. 
The  principles  are  stated  with  clearness  and  precision—  -the  mode  of  reasoning 
is  analytical  and  systematic,  yet  the  character  of  the  work  is  eminently  prac- 
tical, and  well  deserves  the  attention  of  teachers.  We  think  it  cannot  fail  to 
occupy  a  prominent  place  among  the  best  text-books  upou  this  science  HOW 
in  use. 

JOSEPH  M.  KEESE,  President  of  the  New  York  State  Teach- 
ers' Association,  and  Principal  of  Public  School  No.  5. 
We  heartily  concur  in  the  opinion  expressed  above. 

DAVID  PATTERSON,  M.D..  Principal  of  Public  School  No.  3. 
WM.  BELPEN,  Principal  o:  Public  School  No.  2. 
LEONARD  HAZEi/l'lNE,  Principal  of  Public  School  No.  14. 
ABM.  K.  VAN  VLECK,  Principal  of  Public  School  No.  16. 

From  Wm.  Belden,  Jr.,  A.  M.,  Principal  of  Ward  School  No.  3,  New  York  City. 

A  careful  examination  of  Prof.  Thomson's  "Practical  Arithmetic"  has  satis- 
fied me  that  it  is  a  work  of  uncommon  merit. 

The  plan  of  presenting  examples,  in  order  to  introduce  the  rule  by  previously 
analyzing  its  principles,  which  I  consider  the  most  important  distinctive  fea- 
ture of  the  work,  will  commend  itself  to  every  experienced  teacher,  as  the 
natural  process,  both  for  imparting  knowledge  of  this  subject,  and  giving  cor- 
rect habits  of  mental  discipline. 

The  language  of  the  explanations  and  rules  is  peculiarly  clear  and  intelligi- 
ble and  the  amount  and  value  of  this  part  of  the  work  much  superior  to  that  of 
any  other  Arithmetic  with  which  I  am  acquainted.  The  number  and  gradu- 
ally progressive  character  of  the  examples  are  also  worthy  of  special  notice. 

VVM.  BJc<IjL)Jk.N,  Jr«y  A.  JVI. 


New  York,  Sept.  9,  1846. 
I  heartily  concur  in  the  above  recommendation. 

S.  S.  ST.  JOHN,  A.  M.,  Principal  of  Ward  School  No.  10 

From  Solomon  Jsnner,  Esq.,  Principal  of  the  Commercial  and  Classical  School, 

New  York  City. 

To  the  Public.  —  Among  the  numerous  treatises  on  the  science  of  Arithmetic 
which  I  have  carefully  examined,  I  believe  that  Day  and  Thomson's  is  the 
oest  adapted  to  aid  the  teacher  aud  facilitate  the  progress  of  the  learner. 

New  York,  9th  mo.  9th,  1846. 

From  E.  Hotmer,  A.  M.,  Principal  of  Moravia  Institute,  N.  Y. 
Mv  dear  Sir,—  I  have  given  your  •'  Practical  Arithmetic"  a  careful  examiim- 
tton,  and  feel  so  well  a*»ured  ef  1U  swperiwity  over  other  week*  of  the  kind 


which  hav»  fallen  under  my  observation,  that  I  hare  adopted  It  in  ottr  lufl 
lute.  ^t'espectfully  yours, 

E.  IIOSMER 
Moravia,  March  '2- 


,  Principal  of  Norwich  Academy,  N.  Y. 
Mr.  J.  B.  ThniiisoiAyefTf  Sir,— 1  have  examined,  with  much  care  and  in' 
terest,  your  "  Practical,  Ajmhmetic,"  and,  without  attempting  to  specify  its  va- 
rious..excellencies,  1  assure  you,  it  approaches  nearer  to  my  idea  of  a  complete 
BLwith  which  1  am  acquainted.     1  shall  embrace  the 
earliest  opportunity  to  introduce  it  into  the  academy  under  my  charge. 

J.  C.  HOWARD 
Norwich-Apr.iLll.  L&G 

^/Fro,     -  :incip:tlof  East  Bloomfield  Academy,  N.  Y. 

Dear  Sir, — I  have  carefully  examined  your  PRACTICAL  ARITHMETIC.  It  1* 
Jim  such  a  text-book  as  we  want — clear,  concise,  lucid,  logical— ».  Practical 
Arithmetic,  evidently  written  by  a  practical  teacher.  We  shall  introduce  it  a* 
a  text-book  in  this  Institution  at  the  commencement  of  our  next  term. 

Respectfully  yours,  S.  W.  CLARK. 

East  Bloomneld,  May  26,  1846. 

From  R.  M.  Wanzer,  Esq.,  Principal  of  Genoa  Academy,  N.  Y. 

From  the  actual  use  of  Thomson's  "  Practical  Arithmetic"  in  my  School,  1 
am  persuaded  that  it  is  better  adapted  to  give  a  pupil  a  comprehensive  and  prac- 
tical knowledge  ofjiffurtt  than  any  work  of  the  kind  with  which  I  am  acquaint- 
ed. R.  M.  WANZER 

Genoa,  May  17,  1846. 

From  Orson  Barne»,  Esq.,  late  Superintendent  of  Onondaga  Co.,  N.  Y. 
Dear  Sir, — I  have  examined  your  First  Lessons  for  Children ;  also,  your 
Practical  Arithmetic,  and  earnestly  commend  them  to  the  attention  of  parents 
and  teachers.  In  my  estimation  their  introduction  into  our  schools  would 
prove  a  powerful  auxiliary  in  developing  and  disciplining  the  intellectual  fao 
ulties  of  the  pupils,  and  essentially  aid  the  teacher,  in  his  arduous  toils. 

Respectfully  yours,  ORSON  BARNES 

Baldvvinsville,  May  28,  1846. 

From  Wm.  A.  Cropsey,  Esq.,  Town  Superintendent  of  Locke,  N.  Y. 

I  have  examined  Thomson's  "  Practical  Arithmetic,"  and  am  fully  convln 
eed  that  it  is  a  work  of  great  merit.  Its  superiority  will  be  readily  seen  in  the 
excellent  arrangement  of  the  work,  the  clearness  and  simplicity  of  the  rules 
and  explanations,  and  the  well  selected  examples,  which  are  all  admirably 
adapted  to  the  capacities  of  the  young.  I  think  the  arithmetics  now  in  gene- 
ral use  in  this  section,  will  soon  be  laid  aside,  and  the  "  Practical  Arithmetic" 
supply  their  places.  I  would  be  much  pleased  to  see  it  introduced  into  every 
ichool  in  this  town.  Yours  truly,  WM.  A.  CROPSEY. 

Locke,  March  27,  1846. 

From  Thos.  J.  Haswell,  A.  M.,  Prin.  of  Chester  Academy,  N.  Y. 

J.  B.  Thomson,  Esq.— Dear  Sir,— I  have  examined  your  "  Practical  Arith- 
metic" and  hesitate  not  to  say  it  is  the  best  I  have  seen.  Its  arrangement  is 
natural — its  rules  are  concise  and  lucid— its  notes  and  observations  appropriate 
and  valuable — its  examples  abundant  and  happily  selected  ;  well  calculated  t« 
interest  the  learner,  and  lead  him  to  a  thorough  knowledge  of  the  science.  I 
•hall  introduce  it  immediately  into  my  school.  Your  Algebra  1  have  already 
In  use,  and  shall  introduce  your  Legendre,  esteeming  it  as  I  do  all  the  books 
•f your  series  which  I  have  seen,  of  superior  merit. 

Yours  sincerely,  THOMAS  J.  HASWELL. 

Chester,  July  14,  1846. 

From  B.  F.  Evtrson,  Esq.,  Principal  of  Public  School,  Union  Springs,  N.  Y 
J.  B.  Ttouwon  Esq.— D*ai  Sir,— I  have  carefully  examined  your  Practical 


Arithmetic,  and  from  the  clearness  and  precision  of  Its  rates  and  expjanationi 
— its  careful  arrangement — the  copiousness  of  its  exarnpres,  ooth  mdfital  and 
for  the  "  board" — the  appropriateness  of  its  suggestions  and  observations,  and 
Its  easy  plan  of  Induction — I  hesitate  not  to  pronounce^  yours  decidedly  the 
most  appropriate  work  for  our  schools  I  have  ever  noticed,  and  shall  use  mj 
Influence  for  its  general  adoption.  Your*  tru%, 

BENMtfft  P.  EVER8ON 
Union  Springs,  May  28, 1846. 


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